Downloaded by guest on September 29, 2021 a planets Hammond locked Mark tidally on circulation of atmospheric components divergent and rotational The PNAS exoplanets and processes study. atmospheric further and merit sensitive parameters likely planetary are the multiple transport for to heat diver- transport day–night and to rotational heat the components the of gent of contributions relative half The divergent around Jupiter. terrestrial the hot for the that accounts in find and transport case we heat Surprisingly, day–night side. dominates from night circulation transport heat to contribu- to side components the circulation day compute the of decomposition we each hot Helmholtz of processes, tion the the atmospheric and These of studying case side. utility for terrestrial night the the the illustrate both on To Jupiter. for that descends cell drawn and isotropic are side roughly conclusions day single, the a on of overturning ascends form divergent, the com- that takes when and circulation negligible velocities not rotational are with velocities pared divergent work, previous that with show rela- contrast In and we flow. structure total the spatial to their contribution evaluate tive to each com- us out allows Separating divergent circulation. component the overturning and the waves, contains stationary ponent and gaseous comprises jet a component equatorial rotational and the the planet planets, terrestrial both For a Jupiter. of applied hot circulation is simulated technique the and This separated to (divergence-free) decomposition, components. be rotational Helmholtz (vorticity-free) into can a divergent circulation features using the these by splits circulation of which total this each the In that from circulation. show overturning we driven work, thermally station- and planetary-scale waves, circula- jet, 2020) ary equatorial atmospheric 30, superrotating October global a review host for with (received tions likely 2021 18, exoplanets February locked approved and Tidally NJ, Princeton, University, Princeton Bahcall, A. Neta by Edited Kingdom United Oxford, 3PU OX1 Oxford, of University Laboratory, pce rmcnesn ntecl ih ieadlaigto leading and side night cold (12–15). the volatile collapse (rocky) on atmospheric prevent terrestrial to condensing needed a from transport crucial species have on heat be also atmosphere sufficient may with will habitable planet, circulation a which the supporting 11), addition, to (10, In clouds effects. and distribution observable Vertical and species transport (6–9). chemical the circulation affects of the atmosphere the by in determined motion is (3–5). atmo- rotates which the of it sphere, structure temperature as structure the cloud understanding planet on and the depend is composition chemical by which of reflected curve,” retrievals or Accurate “phase around emitted observable clouds light the cir- the affects and This The species, planet. habitability. chemical the heat, and transports stability and culation atmospheres atmospheric their of their observations studying interpreting to vital is ets side day permanent a side. yielding star, night their and to that orbital side always planets same planet’s star These the 2). the present host (1, cause synchronize their to star periods to rotational and and planet enough between close stresses tidally are tidal of which circulation the planets, is these locked of notable most The circulation. E eateto h epyia cecs nvriyo hcg,Ciao L667 and 60637; IL Chicago, Chicago, of University Sciences, Geophysical the of Department nesadn h lblcruaino ial okdplan- locked tidally of circulation global the Understanding u,hv eeldavreyo oe om fatmospheric of the forms novel than of other variety stars a revealed orbiting have planets Sun, are which xoplanets, 01Vl 1 o 3e2022705118 13 No. 118 Vol. 2021 | topei circulation atmospheric a,1 n elT Lewis T. Neil and | emot decomposition Helmholtz b,1,2 ulse ac 2 2021. 22, March Published 2 1 the under Published Submission.y Direct PNAS a is article This interest.y competing no declare authors The analyzed research, paper.y performed the wrote research, and designed data, N.T.L. and M.H. contributions: Author function. a using circulation total (25): components (“divergence-free”) the from circu- total lation out superrotating the divides uniquely separated which and decomposition, Helmholtz be waves, can stationary jet circulation, another. overturning one from isolated study be no can as they understood, how poorly shown is has circulation total com- the these jet of to each equatorial ponents of which (superrotating) contribution relative 21), prograde the (20, However, a (21–24). waves accelerate motion stationary can vertical of This turn generation (16). in the side to night leads the forcing on then day–night sinking and The the side on (16–19). rising day air sides features which night circulation, overturning and between generates gradient sides heating/cooling day strong the their by driven is planets where Above, u rmtowl-tde eea iclto oe (GCM) model circulation general out- well-studied fields two velocity from horizontal the put to decomposition Helmholtz owo orsodnemyb drse.Eal [email protected] Email: addressed. be may correspondence whom To work.y this to equally contributed N.T.L. and M.H. rmtedysd otengtsd,ee hntejtis jet the when even side, night the transport to heat side strong. dominate day different can the to the that due from component reveals transport analysis day–night heat This direct circulation. quantify the to of illustra- it components an use As atmospheres. we locked tion, tidally of study future significantly math- aid simple will a decomposition using This day separated decomposition. planet, be the ematical side—can from the night flow the around to direct going side and waves, jet atmospheric circulation—a com- stationary main the the that of show on ponents We habitability. circulation their and observations assessing telescope atmospheric and interpreting side for crucial the day is planets permanent Understanding these a side. have night exoplanets locked Tidally Significance nti td eadesti su ysoighwthe how showing by issue this address we study this In locked tidally on circulation the that shown has work Previous u b δ topei,OencadPaeayPyis Clarendon Physics, Planetary and Oceanic Atmospheric, χ (u = stedvrec and divergence the is χ stevlct oeta ucin and function, potential velocity the is and , v NSlicense.y PNAS ) ψ nodvret(vriiyfe” n rotational and (“vorticity-free”) divergent into r bandfrom obtained are u https://doi.org/10.1073/pnas.2022705118 = = u −∇χ d ∇ ∇ + 2 2 ψ χ u + ζ r = = k stevriiy eapythe apply We vorticity. the is δ ζ ∇ψ. × , ψ sastream- a is | f12 of 1 [3] [1] [2] [4]

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 1. The global circulation of the idealized tidally locked planets simulated in Exo-FMS and THOR. (Left column) Terrestrial simulation, showing the height and velocity fields at 0.4 bar and the zonal-mean zonal velocity. (Right column) Hot Jupiter simulation, showing the temperature and velocity fields at 0.02 bar and the zonal-mean zonal velocity. Both simulations have the eastward equatorial jet and eastward hot-spot shift typical of tidally locked planets. For both simulations, the substellar point is located at (0◦, 0◦). We show the height field for the terrestrial case and the temperature field for the gaseous case for consistency with the original publications where their overall circulation was analyzed (26, 27). In both atmospheres, the height and temperature fields have the same qualitative structure as the thickness of a hydrostatic layer between two pressure levels is proportional to its temperature (22). This relationship is expressed by the hypsometric equation (28).

simulations of tidally locked atmospheres, one representing a produced by these waves (22–24). Fig. 1, Right column shows terrestrial planet (26) and the other a giant gaseous planet the hot Jupiter. Temperature and velocity fields at 0.02 bar are (a “hot Jupiter”) (27). Details of the numerical procedure used shown in Fig. 1, Top Right, and the zonal-mean zonal velocity to invert Eqs. 3 and 4 are provided in Materials and Methods. is shown in Fig. 1, Bottom Right. This atmosphere has the same The Helmholtz decomposition has been used extensively to study key features as the terrestrial case, despite its much higher tem- the Earth’s atmospheric circulation (25). We apply it for the first perature, larger size, and faster rotation rate. In this study we time to the atmospheric circulation of tidally locked planets. decompose the velocity fields of our two simulations into two Fig. 1 shows the global circulation of the terrestrial planet physically distinct circulations and show how they relate to these and hot Jupiter simulations that we analyze in this study. The key features. terrestrial simulation was run using the GCM Exo-FMS (24, 26), using parameters appropriate for typical terrestrial tidally Results locked planets, such as those in the Trappist-1 system (29). Terrestrial Planet. Fig. 2 shows ur and ud at 0.4 bar, calculated The hot Jupiter simulation was run using the GCM THOR (27, from Eqs. 1–4 for the terrestrial circulation shown in Fig. 1. 30), configured with parameters appropriate for the planet HD The rotational circulation is characterized primarily by zonal 189733b (31). In both simulations, the substellar point is located flow around the planet’s axis of rotation, superimposed over a at 0◦ longitude, 0◦ latitude. Model details and parameters are stationary wave pattern, and the divergent circulation is char- described in Materials and Methods for both cases. The data for acterized by roughly isotropic flow away from the substellar the THOR simulation were provided to us by the developers point. of THOR. The rotational circulation has two phenomenological com- Fig. 1, Left column shows the circulation of the terrestrial ponents, which are manifest in the zonal mean of ur and the planet simulation. Fig. 1, Top Left shows the height and veloc- eddy component of ur (Fig. 3). The eddy component corre- ity field at 0.4 bar, and Fig. 1, Bottom Left shows the zonal-mean sponds to planetary waves, primarily stationary Rossby waves zonal velocity. These fields show the key features of its circu- with zonal wavenumber 1 (22–24), which are driven by the lation: a “hot spot” shifted eastward of the substellar point, divergent circulation (20). The zonal-mean component of ur stationary planetary-scale waves, and an eastward equatorial jet corresponds to the zonal-mean superrotating jet, which is

2 of 12 | PNAS Hammond and Lewis https://doi.org/10.1073/pnas.2022705118 The rotational and divergent components of atmospheric circulation on tidally locked planets Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 dy(oa iu oa en atof part mean) zonal minus (total Eddy 3. Fig. con- of important aspects an any features therefore is include two 3 not these Fig. do that circulation. overturning also and the They flow effects. zonal wavenum- separable zonal-mean zonal have a with is and response atmosphere 1 wave the ber stationary that a assumes by Each models dominated planets. and shallow-water these Hammond on these forms and how of shift explained (23) hot-spot which aforementioned models, al. the the shallow-water on et jet in eastward waves Tsai the of stationary effect way. the sug- included same (24) model, atmospheres Pierrehumbert the (3D) shallow-water three-dimensional in fully the in forms eastward jet in the an equator that gesting produce the waves at These contrast) acceleration waves. heating Polvani day–night stationary spatially and the a produces with (representing Showman system heating layer. shallow-water of a periodic fluid forcing mode single how vertical showed a “shallow-water (22) baroclinic of as first results atmosphere the the the represent on largely which built models,” to been topography has planets surface no is in discontinuity there a produce (assuming longitude is over part is velocity meridional this the because merid- zonal-mean velocity no rotational is There ional 26). (22, waves stationary associated the equator with the toward transport momentum by produced (Right 1. cell. wavenumber overturning zonal single with a wave with stationary associated a be and to jet show equatorial eastward zonal-mean the u, of composed is component this 2. Fig. h oainladdvretcmoet famshrccruaino ial okdplanets locked tidally on circulation atmospheric of components divergent and rotational The Lewis and Hammond u xsigter ftegoa iclto ftdlylocked tidally of circulation global the of theory Existing d ttepesr ee hw ee h iegn opnn sdmntdb niorpcflwaa rmtesbtla on (0 point substellar the from away flow isotropic an by dominated is component divergent the here, shown level pressure the At . h w hsclcmoet fterttoa iclto nFg .(Left 2. Fig. in circulation rotational the of components physical two The emot eopsto fhrzna velocity horizontal of decomposition Helmholtz ∂ψ/∂ ψ ). x hc aihswe integrated when vanishes which u r , u r 0 hc sdmntdb aewt oa aeubr1. wavenumber zonal with wave a by dominated is which , u u r o h ersra iuaina . a.(Left bar. 0.4 at simulation terrestrial the for = k ∇ψ × so , ietwr h uselrpit h ovretrtr o in flow return convergent The point. substellar the toward side starting (positive before side directions, night (886 all the on in subside point to substellar the from the away rises from air away and strongest flow is (negative heating a radiative where primarily (250 point, is substellar atmosphere motion free the horizontal In at divergent 2). quivers Fig. levels: (the in pressure contours those different color three as laid at wind horizontal 4 divergent Fig. 400 The in point. shown by locked substellar is driven tidally the circulation at our overturning heating an in to circulation corresponds divergent simulation phenomenolog- the and, speaking, effects), ically nonhydrostatic (ignoring motion shallow-water a reasonable in is isolation it in that components so system. latter process waves, the stationary distinct model and physically to jet a the is both suggests circulation from also It overturning 1. domi- the wavenumber indeed zonal that with is component a which a to circulation, by corresponds rotational nated response the wave of stationary part their specific that shows models it shallow-water these as in made assumptions the of firmation oa-enpr of part Zonal-mean ) iegn oiotlmto cusdet ovretvertical convergent to due occurs motion horizontal Divergent P,and hPa, P) i eun rmrgoso usdneo h night the on subsidence of regions from returns air hPa), ω n iegs hsflwetnsruhyisotropically roughly extends flow This diverges. and ) 886 P,wt h rsuevria velocity vertical pressure the with hPa, u oainlcmoetof component Rotational ) r , u r hc steeswr qaoiljt (Right jet. equatorial eastward the is which , https://doi.org/10.1073/pnas.2022705118 ω 400 .I h onaylayer boundary the In ). iegn opnn of component Divergent ) P r h aeas same the are hPa u, P and hPa u ◦ 0 , r i.3sosthat shows 3 Fig. . ◦ ,wihw will we which ), PNAS 400 ω 250 | under- f12 of 3 hPa), hPa, )

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 4. The divergent circulation of the terrestrial case. Each panel shows vertical velocity (shaded) and horizontal velocity (quivers). First to third panels: Latitude–longitude slices at 250 hPa (Upper Left), 400 hPa (Upper Right), and 886 hPa (Lower Left), showing the day–night overturning cell. Fourth panel, Lower Right: Slice at 400 hPa shown in tidally locked coordinates (main text), showing how the divergent flow is primarily along lines of constant tidally locked longitude in this coordinate system.

the boundary layer is roughly isotropic, like the divergent flow 37), by analogy with their namesakes in the Earth’s tropical aloft. There is some anisotropy in the boundary layer ascent (i.e., atmosphere. On the night side, meridional motion has been ω) near the substellar point, which takes the form of a wide, described as an “anti-Hadley circulation” (35, 37). Our anal- horseshoe-shaped feature. This feature appears similar in struc- ysis of the divergent circulation suggests that this interpreta- ture to a weather front in the Earth’s atmosphere (32) and forms tion of overturning in a tidally locked atmosphere is overly at the boundary where cold air, advected from the night side by complicated and misleading; the overturning circulation is not the rotational circulation (which is predominantly westward in three different circulations, but instead a single thermally direct the boundary layer), meets the warm air in the vicinity of the and (roughly) isotropic circulation from the day side to the substellar point. night side. At 400 hPa, it is notable that there are two main regions of Given the roughly isotropic nature of the overturning circu- convergence just beyond each terminator, rather than a single lation about the substellar point, it is instructive to visualize region at the antistellar point, suggesting that the overturning it within a coordinate system that exploits symmetry about the circulation is not sufficiently strong to reach all of the way to axis connecting the substellar and antistellar points. We there- the antistellar point in the midtroposphere for the specific ter- fore adopt the “tidally locked coordinates” of Koll and Abbot restrial planet considered here. Higher up in the atmosphere at (38) (Materials and Methods), where tidally locked latitude ϑ0 is 250 hPa, divergent flow extends farther and convergence primar- measured as the angle from the terminator, and tidally locked ily occurs near the antistellar point. The divergent circulation is longitude λ0 is the angle about the axis connecting the substellar slightly stronger going east from the substellar point, compared and antistellar points. Lines at constant tidally locked longitude to its westward part, likely due to the presence of the zonal-mean (i.e., tidally locked meridians) are arcs that connect the sub- eastward equatorial jet; notably, this effect is more pronounced stellar and antistellar points. Tidally locked velocities u0 and v 0 at 400 hPa where the jet is stronger. are defined in the λ0 and ϑ0 directions, respectively. The point Past interpretation of the overturning circulation on tidally (ϑ0, λ0) = (0, 0) is chosen to coincide with the north pole. We locked planets has described the circulation as a superposi- note that the Helmholtz decomposition (Eqs. 1–4) of the hori- tion of various different overturning circulations. Ascent on zontal velocities is invariant under the coordinate system rotation the day side paired with meridional horizontal motion and from latitude–longitude to tidally locked coordinates. This fol- descent near the poles has been described as a sort of Hadley lows from the invariance of the ∇2 operator under rotation and circulation (33–37), and overturning circulation in the zonal the fact that as ζ and δ are scalar fields, they are also invariant direction has been described as a Walker circulation (36, under rotation by definition.

4 of 12 | PNAS Hammond and Lewis https://doi.org/10.1073/pnas.2022705118 The rotational and divergent components of atmospheric circulation on tidally locked planets Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h oainladdvretcmoet famshrccruaino ial okdplanets locked tidally on circulation atmospheric of components divergent and rotational The Lewis and substellar Hammond the toward back flow return point. near-surface (ϑ (ϑ a terminator point the substellar enters across the passes near air rises as features Air the 4. of Fig. 90 each in reveals shown and the were side from inferred night circulation the the of to side day the the pressure particular the a in above flux flux mass the point) (in stellar northward total the is where hc nbe h ento fatdlylce eiinlmass meridional locked tidally a of streamfunction definition the enables which the in Symmetry 5. Fig. oriae,tecniut qaini ial locked tidally in equation continuity is coordinates the coordinates, in symmetry zonal approximate con- of of form the lines in along coordinate flow this “southward” in of stant clear form is the flow taking dom- antistellar system, The to (i.e., motion. substellar vertical terminator of descending the is is inance there there of side), side), night “south” the day and on termi- the motion, the on (i.e., vertical is terminator ascending system the coordinate of Consequently, locked “North” at system. nator. tidally now coordinate the coordinate, is new in motion latitude–longitude “equator” the vertical the ascending in substellar in “pole” all a equator now the is on was which point locked point, substellar tidally a The to data. the moving transforms how coordinates illustrates coordinates, longitude Right Upper at system coordinate locked tidally the between magnitude in differs scale color the that Note motion. by anticlockwise captured flux indicate mass values maximum negative The and panels. motion, two clockwise indicate values Positive streamfunction ional Ψ yaaoywt h otniyeuto nregular in equation continuity the with analogy By 4, Fig. ◦ ,bfr oigtwr h ih ieaot ecn begins Descent aloft. side night the toward moving before ), ϑ 0 λ 0 ssoni i.5 h vrunn iclto iwdin viewed circulation overturning The 5. Fig. in shown is –p [ 0 h vrunn iclto ftetretilpae.(Left planet. terrestrial the of circulation overturning The ·] h stoi aueo h iegn iclto appears circulation divergent the of nature isotropic The . a λ oe Right Lower ln ae h omo igeoetrigcl from cell overturning single a of form the takes plane cos 0 ϑ 1 niae naeaeoe ial okdlongitude. locked tidally over average an indicates 0 ae nFg ,wihshows which 4, Fig. in panel –p ϑ 0 ln olw ie fconstant of lines follows plane ∂ ∂λ λ Ψ u Ψ 0 0 0 0 ieto oiae nertn Eq. integrating motivates direction = asflxfloslnso constant of lines follows flux Mass . + ae hw h iegn velocity divergent the shows panel 2π a cos a 1 g cos ϑ 0 ϑ ϑ ∂ϑ 0 0 ∂ ieto,ie,twr h anti- the toward i.e., direction, Z 0 0 0.4 λ p v –ϑ  0 0 v a.Cmaio ihthe with Comparison bar. cos 0 = 0 lcsof slices  u λ Ψ ◦ ϑ d 0 0 dp ,adeetal air eventually and ), 0 smc rae hnta mle by implied that than greater much is nrglrlatitude– regular in
+ λ , Ψ 0 0 . ∂ω ∂ u . p d ϑ Ψ and 0 p 0 0 = 90 = or n mass and , 5 ial okdstreamfunction locked Tidally ) u h ieto foetrigi niae yarw n h ino h streamfunction. the of sign the and arrows by indicated is overturning of direction The Ψ. . ω over d ◦ which nthe in The . λ–ϑ [6] [5] 0 λ Ψ = 0 0 , lnt.Fg hw h eopsto ftettlvelocity total the of these decomposition of the typical shift shows hot-spot 6 and Fig. jet planets. equatorial terrestrial eastward reveals the the it in that were show that and (31) components simulation. 189733b circulation HD same Jupiter the hot we the (Eqs. section, is this of decomposition clouds In Helmholtz and 5). the (4, species, observations apply its chemical these heat, and interpreting of to circulation vital transport global the on mea- their or temperatures effect Understanding curves high phase spectra. and observing orbits, for suring targets short excellent sizes, them large make above temperatures Their stars, and host K. days their few 1,000 a to of close periods orbital orbit them They giving Jupiter. to comparable radii Jupiter. Hot to returning and point. side substellar night roughly the the diverges on and subsiding point before substellar isotropically, the at The rises waves). that stationary overturning circulation direct (the thermally eddies to corresponds and circulation jet) divergent (the from contributions mean into zonal circulation separated the rotational further the be atmo- dividing can the by features of out part two stationary These the circulation. and by spheric circulation produced divergent jet the the zonal-mean of by the composed forced circu- waves primarily meaningful Rossby is stationary physically circulation two rotational components The into divergent lations. division and a to rotational corresponds into atmosphere locked How- as 37). longitudes. (35, motivate, all over side) to equation night continuity difficult or the is integrating side approach day this the ever, just in over described (i.e., been range where have dif- works circulation day–night overturning that previous note the We in motion. ferences meridional cross-polar streamfunction or regular The fluxes. Ψ mass meridional 33– night-side by (27, authors indicated by previous that indicated regu- by strength circulation considered The the 5), 39) 35, Fig. than in better shown (also circulation streamfunction mass the meridional of lar nature the captures i.1sostegoa iclto fti o uie,with Jupiter, hot this of circulation global the shows 1 Fig. tidally terrestrial a of circulation the dividing summary, In iulzn h vrunn iclto ntrsof terms in circulation overturning the Visualizing locpue oeo h vrunn nteznldirection zonal the in overturning the of none captures also niaigthat indicating Ψ, o uiesaegsostdlylce lnt with planets locked tidally gaseous are Jupiters Hot Ψ 0 endb Eq. by defined Ψ 0 u ocneainbtendysd and day-side between cancelation to due Ψ Ψ 0 saeae vralmtdlongitude limited a over averaged is atrsmr fteoetrigthan overturning the of more captures (Right 4. https://doi.org/10.1073/pnas.2022705118 2π = Ψ eua ueinma merid- mean Eulerian Regular ) oasimulation a to 1–4) Ψ a cos sfrwae than weaker far is Ψ ϑ PNAS sdrvdby derived is R p 0 Ψ [v 0 | ] λ clearly f12 of 5 dp Ψ. /g

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 6. Helmholtz decomposition of horizontal velocity u for the hot Jupiter at 0.02 bar. (Left) Rotational component of u, ur, showing the eastward jet ◦ ◦ with a relatively weak stationary wave. (Right) Divergent component of u, ud, dominated by a divergent flow away from the substellar point (0 , 0 ) which is less isotropic than the equivalent flow in the terrestrial simulation, which we suggest is due to the relatively stronger equatorial jet in the hot Jupiter atmosphere.

field into the rotational and divergent velocity components ur the simulation means that this cannot be due to the linear Kelvin and ud , at the pressure level 0.02 bar, and Fig. 7 shows the waves demonstrated by Showman and Polvani (22) to produce further decomposition of the rotational component into contri- this tilt and acceleration. Hammond et al. (26) showed how non- butions from the zonal mean and eddies. As with the terrestrial linear equatorial waves can produce this acceleration without an case, the Helmholtz decomposition shows that the total circu- imposed linear drag, which we suggest is the process occurring in lation is the sum of a divergent flow from day side to night these GCM simulations. side, plus a rotational flow dominated by a zonal-mean zonal jet The tidally locked meridional mass streamfunction Ψ0 (Eq. 6) and a wavenumber-1 stationary wave pattern. The divergent flow is shown in Fig. 8 to elucidate the vertical structure of the over- away from the substellar point is roughly isotropic in the north, turning circulation in the atmosphere of the hot Jupiter. Similar south, and eastward directions, but interaction with the zonal jet to the terrestrial atmosphere, the main overturning circulation appears to prevent the divergent circulation from extending very takes the form of a thermally direct cell rising on the day side and far in the westward direction against the flow of the jet. The eddy sinking on the night side just beyond the terminator. In addition component of the rotational flow is difficult to see in Fig. 6, as it to the main “clockwise” overturning cell, there is a weaker “anti- is weaker than the jet, and it is more clearly appreciated in Fig. clockwise” cell on the night side. At first glance, this circulation 7. While the divergent circulation and eddy rotational circulation would appear to be thermally indirect. However, there is a local are weaker than the jet, neither one is negligibly small, with char- temperature maximum around ϑ0 = −90◦, p = 1 bar (see, e.g., acteristic velocities only a single order of magnitude less than the the equatorial temperature profiles in figure 5 of ref. 40), gen- jet speed. erated by heat deposition by the extended “tongue” of the main The eddy rotational component is dominated by a stationary clockwise circulation, which may drive the anticlockwise circula- wave response with zonal wavenumber 1, as it was in the ter- tion making it thermally direct. Overturning motion in the hot restrial case, matching the assumptions of previous studies using Jupiter atmosphere is restricted to pressures less than roughly shallow-water models (22–24). It also shows a clear “tilt” simi- 3 bar. This is because there is little shortwave heating below this lar to that suggested by Showman and Polvani (22) to accelerate level to drive the circulation, as the shortwave optical depth is the equatorial jet, although the lack of a linear Rayleigh drag in roughly 10 at 3 bar (see Table 1). Below 3 bar the divergent

Fig. 7. The two main parts of the rotational circulation shown in Fig. 6, decomposed as in Fig. 3. (Left) Zonal-mean part of ur, ur, showing the equatorial 0 jet. (Right) Eddy (total minus zonal mean) part of ur, ur , showing a wavenumber-1 wave that is weaker than the equatorial jet.

6 of 12 | PNAS Hammond and Lewis https://doi.org/10.1073/pnas.2022705118 The rotational and divergent components of atmospheric circulation on tidally locked planets Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h oainladdvretcmoet famshrccruaino ial okdplanets locked tidally on circulation atmospheric of components divergent and rotational The Lewis and Hammond terrestrial locked tidally rotating slowly planets. of fea- that circulation important the suggested an of has only ture is which overturning 44–46) driven 39, thermally 38, day–night contrast (33, in work fea- is previous ubiquitous This a circulations. with is atmospheric divergent side locked night tidally a the of that to ture side indicates day the similarity from This circulation of those planet. from terrestrial different very the on—are so as rate, and notable, rotation temperature, is composition, radius, This Jupiter—its hot strongest. the is similar of jet very parameters the the is where components level two pressure these the at of day–night structure the single that shows the of which portion 43) night-side 42, 35, the “anti- cell. 27, overturning an just (10, of side actually existence night the is the implies the on and circulation” of flux the Hadley misses mass metric it the good as of planets, a majority locked streamfunction tidally in not a on cell circulation is case, overturning terrestrial overturning coordinates the the latitude–longitude for of in cancelation As the portions average. to zonal night-side due the and mostly is day-side This of simulation. over- the of in than majority turning the capture weaker not does times streamfunction traditional 20 approximately is that as such flow mean residual side over- 3D the a study 41. of to ref. Helmholtz in interest terms outlined the of facilitated using in be side, circulation also turning night would divergent the It the on decomposition. substellar of overturning the isolation the the examine by of of could west feature structure this the 3D investigating to work Future concentrated anticlock- point. is weaker the circulation isotropic, wise roughly is a circulation clockwise above approximately concentrated of of longitude, regions multiple pressure in (11) the show downwelling al. which with and et drag, consistent upwelling Komacek weak is in with features simulations shown circulation these for slices of averaged velocity Each vertical zonally 1). equatorial the Fig. is in (as shown quiescent is circulation case, terrestrial the overturning. in the As of motion. more anticlockwise much indicate captures values it negative that and motion, clockwise indicate values Positive streamfunction o eietfidi the in even can- identified that an midlatitudes be and the circulation” not in circulation “anti-Hadley regular direct weak thermally the implies a weaker longitude) of consequently over existence and averaging the symmetry, by (obtained zonal streamfunction little very has 8. Fig. Comparing streamfunction mass meridional regular The Ψ h vrunn iclto ftehtJptr (Left Jupiter. hot the of circulation overturning The 0 nFg o oprsn h iegn circulation divergent The comparison. for 8 Fig. in asflxfloslnso constant of lines follows flux Mass Ψ. u d and u r o h ersra n o uie cases Jupiter hot and terrestrial the for λ − 1 a.W oeta hl h main the while that note We bar. ϑ lcsof slices u Ψ d 0 hw nFg 6. Fig. in shown hwn htthe that showing , Ψ Ψ 0 ssonalong- shown is or ial okdstreamfunction locked Tidally ) h ieto foetrigi niae yarw n h ino h streamfunction. the of sign the and arrows by indicated is overturning of direction The Ψ. Ψ inlcruaini em faslt eoiy e ttransports that it shows yet rota- 2 velocity, the absolute Fig. than of as terms weaker result, to in significantly circulation surprising side is tional night a circulation the is divergent from This the (back side). circulation day and rotational the circulation, the rotational of the part side, of night part the circulation to divergent jet transport the the by Of is side. 96% night the to transported eiul(9 u ehv hsnt rsn h a aafor data raw Wm the present this to for chosen correction latitude– have a a we introduce simplicity. to to but regridding possible (49) there the is residual as by It zero, introduced grid. to residual longitude sum con- small exactly into a not the is it of does components budget decompose divergent This not and circulation. did rotational energy the static but from moist planets, tributions the Sergeev locked for and planet budgets tidally (33) terrestrial similar for Schneider the calculated and (48) for Merlis al. 9, Jupiter. et Fig. hot in the for terms and the plot Eq. and of latitude) average meridional and a flux take stellar and the from surface, atmosphere the the into coming flux net the where and, flux, energy static is dry budget the energy local of the divergence integrated, column to due is heating ature, where is which energy, static dry side side. day divergent the night we and from the transport to rotational section, heat to the this circulation the of In of components contribution (47). relative curves the its phase and assess to (12–15) thermal due stability atmospheres, of climate these and observations of collapse atmospheric feature This on key circulation. effect a atmospheric produces is the naturally transport by planets heat transport locked heat tidally day–night of a sides night and day Transport. 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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Fig. 9. The dry static energy budget for the terrestrial planet and hot Jupiter. (Left) Terrestrial planet, showing that the day–night heat transport is dominated by the divergent (overturning) circulation, and the jet has little effect. (Right) Hot Jupiter, showing that the jet and divergent circulation contribute roughly equally to day–night heat transport. The net effect of the rotational circulation on day–night heat transport is small, however, as the eddy rotational component transports heat from the night side to the day side, counteracting the jet.

far more energy to the night side. Divergent heat transport is trial planet, as it cannot extend very far away from the substellar roughly symmetric in the eastward and westward directions in point to the west, due to interaction with the stronger zonal the terrestrial simulation, reflecting the isotropic nature of the mean jet (note that divergent heat transport is not symmetric divergent circulation. about 0◦ longitude, unlike in the terrestrial atmosphere). Both of The dominance of the divergent circulation in this case can be these features lead to stronger day–night gradients in s than are explained by the fact that the atmosphere is in a “weak temper- found in the terrestrial atmosphere, which enhances rotational ature gradient” (WTG) regime (18, 50). In this regime, which is heat transport in the hot Jupiter atmosphere by making the sec- typical of planets that are slowly rotating, the zonal momentum ond term in Eq. 9 larger. This leads to enhanced heat transport equation is dominated by a balance between the local tempera- from the day side to the night side by the superrotating jet, as the ture gradient and a term that depends on the square of the local jet flows through a negative gradient in s (temperature decreas- velocity (rather than the Coriolis force) (26). This means that the ing relative to flow direction) downstream of the substellar point. temperature gradients are relatively small. Expanding the local The eddy rotational circulation flows back toward the substellar heating due to the divergence of dry static energy flux point near the equator (Fig. 7), so partially counteracts day–night heat transport by the jet by returning heat to the day side. The net ∇ · su = s∇ · u + u · ∇s [9] effect is that the majority of day–night heat transport on the hot Jupiter is due to the divergent circulation. Unlike the terrestrial case, the rotational circulation also provides a first-order contri- shows that the contribution of the rotational velocity ur will only bution to the heat budget due to the larger gradients in dry static be through the second term on the right-hand side, as ∇ · ur = 0. However, the horizontal gradient of the dry static energy ∇s is energy on the hot Jupiter. small in the WTG regime, so the heating due to the u · ∇s terms The fact that the majority of day–night heat transport is due to is small, meaning that the total heating due to the rotational cir- the divergent circulation, for both the terrestrial planet and the hot Jupiter, is an important result, as some previous work study- culation is small. On the other hand, the divergent velocity ud has nonzero divergence so contributes to the local heating via both ing day–night heat transport on tidally locked planets has focused terms on the right-hand side of Eq. 9. This means that the diver- entirely on the role of the jet and stationary waves (43, 51, 52). gent circulation plays a dominant role in the energy budget, as The red line in Fig. 9 (minus the known instellation) corresponds to the broadband thermal phase curve that could be observed for s∇ · ud is much greater than ur · ∇s (or ud · ∇s). For the hot Jupiter, the mean day-side instellation is 119,375 the planet (5), and consequently the energy budget can be con- Wm−2, of which 70% is reradiated from the day side and 35% is strained by observations. Note that the line in Fig. 9 would be transported to the night side by the total circulation. Note that flipped horizontally when viewed as a phase curve, as it is plotted the total residual is larger in the calculation of the dry static as a function of longitude rather than orbital phase here. Phase energy budget of the hot Jupiter than for the terrestrial case, so curves observed from hot Jupiters are already available and will these percentages do not add up to exactly 100%. Seventy-eight increase in quality in the coming years (53). Similar observations percent of the day–night heat transport is by the divergent cir- should become available for terrestrial tidally locked planets culation and 22% by the rotational circulation. The rotational when next-generation telescopes such as the James Webb Space circulation makes only a small contribution because its two parts Telescope come online (38, 54). Separating the effects of the almost cancel each other out. The zonal-mean jet transports rotational and divergent circulations to heat transport on tidally 122% of the eventual net day–night transport to the night side, locked planets over a broad range of parameters could be crucial but the eddy part of the rotational circulation moves −100% of to interpreting phase curve observations. the net transport back toward the day side. The near cancela- tion of these two processes results in the rotational circulation Discussion contributing only 22% of the total day–night heat transport. Summary. In this work, we have applied a Helmholtz decompo- The hot Jupiter is less firmly in the WTG regime as it is more sition (Eq. 1) to the circulation of two well-studied benchmark strongly forced, rotating faster, and larger than the terrestrial simulations of tidally locked planets. We considered the case of planet (18). Additionally, the divergent circulation on the hot a terrestrial planet using the Exo-FMS GCM (24, 26), designed Jupiter is less effective at transporting heat than on the terres- to model planets similar to those in the Trappist-1 system (29),

8 of 12 | PNAS Hammond and Lewis https://doi.org/10.1073/pnas.2022705118 The rotational and divergent components of atmospheric circulation on tidally locked planets Downloaded by guest on September 29, 2021 Downloaded by guest on September 29, 2021 h oainladdvretcmoet famshrccruaino ial okdplanets locked tidally on circulation atmospheric of components divergent and rotational The Lewis and Hammond h ersra ae hsi eas hs lnt r typically are planets these because For components. is other this the case, than terrestrial weaker when the is even velocity transport absolute heat its day–night the dominates overturning or flow cross-polar direction. about zonal the information in no overturning over- the contains substellar of and tradi- most the turning capture the to near contrast, fails in By coordinates defined rising streamfunction latitude–longitude side. mass air night meridional mean features the Eulerian tional on which substellar descending cell and the point dominated single from is a circulation away overturning motion by the as that isotropic (38), reveals This coordinates by point. locked dominated tidally is in it streamfunction mass and a by useful not is planets. it “Walker” locked and tidally suggest for “Hadley” circulations we distinct define mind, meridional to or misleading little in potentially zonal is this the there With scale in that larger directions. overturning means much the isotropy a between its on distinction and occurs scale), cir- it planetary Walker However, (the and Hadley Earth. the on of combination culations similar a is to with respects circulation associated some overturning those in The than circulation. less velocities magnitude rotational with of the negligible, order not single is a than it only weaker but is circulation, circulation rotational studies divergent the previous the cases, by both analyzed For waves (22–24). shallow-water corre- wave the stationary to The sponds circulation. component divergent overturning the the and contains rotational), wavenumber-1 (eddy and circula- wave rotational) rotational stationary (zonal-mean The jet components. the three contains these tion qualitative of the each shows of 10 stationary form Fig. jet, circulation. that superrotating overturning components of and circulation—the main waves, total circulation three the the up atmospheric isolates make planets the locked to tidally decomposition configured Helmholtz 30), (31). (27, 189733b HD model planet THOR the of the parameters using the with Jupiter hot a and locked tidally the of it. use analyze the to motivates sub- system which the coordinate points, connecting the antistellar axis divergent the and the into The on around stellar (green). divided symmetry waves depends is approximate stationary that has eddy circulation circulation the location rotational and a (red) The jet at terminator, circulation. zonal-mean side the the the night of across at the strength isotropically rises on (blue) roughly descending circulation extends before overturning and Divergent, point study. substellar this in tified 10. Fig. hr,tedvretcruainascae ihteday–night the with associated circulation divergent the Third, naturally represented is circulation overturning the Second, a applying First, conclusions. main three has work Our (Rotational) Waves Waves ceai hwn h he ancruaincmoet iden- components circulation main three the showing schematic A

Terminator Overturning (Divergent) Substellar Point (Rotational) Zonal Jet al sflfrtesol oaigtretilcs 3) hr the where (38), case terrestrial rotating particu- slowly be the would the by for model useful a influenced from Such larly weakly rotation. only planet’s directions the is of it all effect cir- the that in means divergent which start- the on isotropic point that good substellar circulation approximately shown overturning a have is the be we culation of as would model planets, (59) simple locked Sobel tidally a for and point Bretherton ing such of circulation Walker that Earth’s ther- the as of of amplitude Models the curves. phase of heat mal interpretation the day–night aid describe would of and predictions that seen transport allow models shift would Developing hot-spot circulation ana- curves. the divergent we predicting phase that with thermal of circulation helps in is speed rotational This which the the above. planets, predict of lyzed locked to part tidally model zonal-mean terrestrial the on simple jet a equatorial different developed the the (26) of Hammond al. models identified. have simple et we testing that circulation for the of basis components a be also would in observations curve phase trans- regimes. thermal for parameter shaping different important and most would are heat which processes other, porting which the ascertain dominate to will help one important is when planets determining locked to tidally of com- classes circulation different divergent the on and ponents determines rotational obvi- the what an of Understanding strength is work. relative transport this to heat follow-up and Helmholtz ous the circulation atmo- on the model moisture of the of in decomposition effect condensate the (i.e., cloud examining dry or and is vapor sphere), work water present no the is terres- in there The analyzed (33). have heating we latent simulation of trial effect the their and both effects through radiative transport, influence heat likely and will dynamics Additionally, clouds atmospheric and (58). the vapor circulation water planets, been atmospheric habitable has the on surface) affect land ocean ter- to a homogeneous of On a shown absence to timescale. or opposed show presence radiative (as the surface we shorter planets, Jupiter its locked tidally hot to restrial suggests the due (57) than the possibly Komacek circulation of above, and divergent Inspection Tan stronger (56). in a pressure simulated timescales boundary radiative Jupiter lower and ultrahot drag the imposed and to been con- (55) has sensitive model Jupiter be hot in Jupiters to a made hot of shown choices circulation and to the example, sensitive planets For be figuration. terrestrial to shown both been on have circulation simulations the addition, In of mass. planetary and instellation, rate, the rotation as radius, such com- parameters circulation planetary each on a of depends strength be ponent Such the space. circu- should how parameter investigate global components wide could study a the divergent spanning of and simulations to division rotational applied the its tidally future, into of the lation simulations In benchmark planets. con- have “exemplar” locked we two study this only in First, sidered study. this by motivated research Work. Future velocities others. higher the much has than component of one circulation, where each whole regimes the in that even to both illustrates important Jupiter is on components hot budget circulation the the heat and day–night planet of the terrestrial dominance the in The circulation still transport. divergent heat circulation the to day–night divergent lead net the transports the that opposing dominates meaning these cancelation, side; day large to from the back a side to heat day transport side the waves night stationary from the the heat but transports side, opposite night jet are the The component directions another. signif- each the one a with However, from make associated budget. do transport heat heat waves For the of stationary waves. to and stationary contribution jet and icant the jet Jupiter, superrotating hot the advec- the by inhibits heat which regime, of gradient” tion temperature “weak a in eod td falreprmtrsaeo simulations of space parameter large a of study a Second, enwotieafwopruiisfrfuture for opportunities few a outline now We https://doi.org/10.1073/pnas.2022705118 PNAS | f12 of 9

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES day–night heat transport is dominated by the divergent circula- pressure coordinate (26). These equations are solved for a dry atmosphere tion so that a model of the divergent circulation could predict the with no representation of water vapor or cloud condensate. Exo-FMS total heat transport and the resulting broadband thermal phase was built on the cubed-sphere dynamical core of the “flexible modeling curve. When the jet, stationary waves, and overturning circula- system” (67). tion are comparably important to the heat budget (as was the The simulation of a terrestrial planet in Exo-FMS was configured in exactly the same way as the simulation in ref. 26 with instellation 1,000 case for the hot Jupiter), one could fit a phase curve to a sum Wm−2. It uses semigray two-stream radiative transfer and dry convective of three basis functions which represent the shape of each com- adjustment (as in refs. 24 and 52), with atmospheric parameters listed in ponent. Phase curves have previously been fitted to a sum of Table 1. We chose to use an albedo of zero to simplify the number of param- spherical harmonics (60, 61) that do not correspond to physical eters chosen, as the value of the albedo is degenerate with the strength of components of the circulation. the instellation. Each of the six faces of the cubed-sphere grid has 48 by 48 Finally, understanding the strength of the divergent circula- cells, which corresponds approximately to a latitude–longitude resolution tion (which accounts for all vertical motion) is vital to models of 1.9◦. The model has 48 vertical levels (this is the only difference from the and observations of disequilibrium chemistry and cloud forma- simulation in ref. 26). The model is stabilized with a fourth-order hyperdif- fusion with a coefficient the same as the default value used for Earth-like tion. These models often rely on a “Kzz ” parameter that describes simulations distributed with this cubed-sphere dynamical core (67). A lin- the strength of diffusive mixing (a parameterization for vertical ear drag is applied to the velocities at the surface with a timescale of 1 d, advection); deriving an estimate of this from a model of the diver- which decreases in strength linearly with pressure up to a value of zero at gent circulation would constrain an aspect of these models that 700 mbar. The test was spun up for 1,000 (Earth) days, until the total angu- is not well understood (see ref. 62 for various ways to estimate lar momentum and total outgoing longwave radiation were equilibrated, this parameter). Modeling the transport due to the rotational and the output data were taken as an average over the next 1,000 d. Its and divergent circulations of atmospheric tracers would also be orbital period of 10 d is an approximate representation of a tidally locked valuable to understanding the distribution of clouds (63), the dis- Earth-sized and Earth-temperature planet orbiting an M dwarf, similar to tribution of chemical species in disequilibrium models (64), and the Trappist-1 planets (29). the heating due to recombination of dissociated hydrogen (65). Hot Jupiter. The simulation of the hot Jupiter was run in THOR, which solves This study has shown how the global circulation of tidally the nonhydrostatic Euler equations on an icosahedral grid with a vertical locked planets can be divided into three physically meaningful height coordinate (27). The “deep” form of these equations is solved, mean- components, summarized in Fig. 10. Each of these components ing that the vertical coordinate z is always used instead of the radius r, which plays an important role, even when they are not immediately is substituted in some parts of the “shallow” equations used in the simula- apparent in the total circulation. We hope that this will form tion of the terrestrial planet in FMS. We are grateful to Russell Deitrick and a basis for future work on the behavior of each of these the rest of the THOR development team who provided us with the data for components and their role in the global circulation. this simulation. The simulation of the hot Jupiter is exactly the same as the nonhydro- static benchmark of HD 189733b presented in ref. 27. This simulation uses Materials and Methods semigray radiative transfer and dry convective adjustment, just like the Description of General Circulation Models. The terrestrial simulation was run simulation in Exo-FMS, with parameters listed in Table 1. The icosahedral ◦ using “Exo-FMS” (26), and the hot Jupiter simulation was run using “THOR” grid has a horizontal resolution corresponding approximately to 2 and (27). We chose to use these existing simulations as benchmarks, as they have 40 vertical levels. The model is stabilized with a fourth-order hyperdiffu- already had their basic circulations analyzed in refs. 26 and 27; this allows sion as discussed in ref. 27, as well as a sponge layer applying a weak us to progress directly to analyzing the circulation features revealed by the linear drag to the eddy part of the zonal velocity near the top of the Helmholtz decomposition. atmosphere. The simulation was spun up for 9,000 d and then output data were recorded as an average over the next 1,000 d. Ref. 27 discusses the Terrestrial Planet. The terrestrial simulation was run in Exo-FMS, which spin-up of this simulation. Their figure 29 shows that the total angular solves the primitive equations on a cubed-sphere grid with a vertical sigma- momentum has equilibrated after 5,000 d. This is a typical hot Jupiter that

Table 1. Parameters of the terrestrial and gaseous simulations in the models Exo-FMS and THOR Terrestrial (FMS) Gaseous (THOR) Earth Jupiter Planetary parameters Radius a (m) 6.371 × 106 7.970 × 107 6.371 × 106 7.149 × 107 Rotation rate Ω (s−1) 7.29 × 10−6 3.279 × 10−5 7.29 × 10−5 1.76 × 10−4 Gravitational acceleration g (ms−2) 9.81 21.4 9.81 24.79 5 7 5 Lower boundary pressure p0 (Pa) 10 2.20 × 10 10 — Gas constant R (JK−1·kg−1) 287 3,779 287 3,745 −1 −1 Heat capacity cp (JK ·kg ) 1,005 13,226.8 1,005 12,359.1 −2 Instellation FS (Wm ) 1,000 467,072 1,361 50.26 Radiative parameters

Albedo A0 0 0.18 — — Longwave optical depth scaling nLW 1 2 — — Longwave optical depth τLW 1 4,680 — — Shortwave optical depth scaling nSW — 1 — — Shortwave optical depth τSW 0 1,170 — — Approximate nondimensional length scales

Equatorial deformation scale Leq/a 1.77 0.68 0.3 0.03 Rhines scale LR/a 0.63 0.82 0.5 0.1

Both models use semigray radiative transfer schemes and dry convective adjustment. Refs. 24 and 27 discuss the respective models and simulations in more detail. We have included columns with the parameters of Earth and Jupiter for comparison, for which we have quoted the nondimensional length scales estimated in ref. 66.

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Wordsworth, D. R. Showman, P. A. 16. 9 .Gillon M. 29. rnfre notdlylce oriae ytetransformation the by coordinates locked tidally into point. tidally transformed antistellar constant the of to point lines substellar system, the coordinate from run this longitude In locked point. antistellar the be axis. substellar–antistellar (ϑ, the about choose angle They the be to longitude define locked They latitude tidally point. antistellar locked the to tidally are point the the coordinates substellar that the The from so runs planet. coordinates, axis latitude–longitude locked polar regular tidally of a rotation a of effectively atmosphere the in present oaigot eua rd h etrhrzna velocity horizontal vector The grid. inter- into regular transformed then a and coordinates onto locked polating tidally into locations their transforming (e.g., fields Scalar osi rmteEgihSekn no.NTL a upre yScience Fel- by Trust ST/S505638/1. supported Lindemann Grant was Council a early N.T.L. Facilities by Technology an Union. and supported English-Speaking on was the comments Komacek, M.H. from useful Tad lowship manuscript. providing to this for grateful of Read are trans- version Peter We to and coordinates. used Abbot, code locked the Dorian tidally model sharing their to for the making data Koll for thank sharing and Daniel form for We us thank GCM with We throughout. THOR 189733b available. the HD work publicly of of this simulation developers the the refine of of results to all us and Deitrick helped Russell be that referees can mous and available, ACKNOWLEDGMENTS. publicly is code model (https://github.com/exoclime/THOR). GitHub available THOR from obtained is The Uni- data request. simulation Oxford upon THOR at The available (72). are (https://ora.ox.ac.uk/objects/uuid:1a32e3b6- c656-414f-81c4-d1ec1f929f4e) manuscript (ORA) this Archive for Research versity analysis data conduct to Availability. Data provided code a at using Koll transformations D. these by calculate We 38. ref. in derived oladAbt(8 hwta aiuelniuecodntscnbe can coordinates latitude–longitude that show (38) Abbot and Koll iclto model. circulation planets. Terrestrial I. planets. locked planets. locked Motion tidally of (2018). circulation Jupiter. hot atmospheric locked tidally a of atmosphere the J. in Astrophys. super-rotation resulting the and J. Astrophys. Lett. Res. Geophys. divergence. tropical idealized Mech. Fluid Rev. Annu. Sci. Planet Arizona 277–326. of pp. (University 1675, Eds. vol. Bullock, A. 2013), M. AZ, Harder, Tuscon, W. Press, J. Simon-Miller, A. in Mackwell, J. exoplanets” S. terrestrial of tion wr trTrappist-1. star dwarf 92. vol. 2006), Astrophys. Astron. MGa-il e ok Y 1976). NY, York, New (McGraw-Hill, ee eprt ersra lnt rudtenab ultracool nearby the around planets terrestrial temperate Seven al., et 0–4 (2015). 509–540 43, https://github.com/ddbkoll/tidally-locked-coordinates. HR20 ao mrvmnst h pnsuc general open-source the to improvements Major 2.0: THOR al., et 4 (2014). 141 793, (2011). 71 738, h esls id nItouto oteTer fAtmospheric of Theory the to Introduction An Wind. Ceaseless The λ) u h x-M iuaindt n h yhncd used code python the and data simulation Exo-FMS The ω = srpy.J Suppl. J. Astrophys. 0 9 (2020). 099 20, 181(2010). L18811 37, nlttd–ogtd oriae r rnfre by transformed are coordinates latitude–longitude in ) = u v 0 )t etesbtla on and point substellar the be to 0) (0, Nature 0 0 (u 7–0 (2019). 275–303 51, = = eaegaeu o h omnso w anony- two of comments the for grateful are We 0 , cos ∂ϑ v ∂λ ϑ λ topei cec:A nrdcoySurvey Introductory An Science: Atmospheric 0 ϑ 0 0 5–6 (2017). 456–460 542, sn h relation the using ) .Ams Sci. Atmos. J. 0 ϑ 0 = = cos 0 ob h nl otetriao n the and terminator the to angle the be to tan sin  u oprtv lmtlg fTretilPlanets, Terrestrial of Climatology Comparative ϑ ∂λ srpy.J. Astrophys. −1 ∂λ −1 0(2020). 30 248, + 0 (cos https://doi.org/10.1073/pnas.2022705118  cos ∂ϑ ∂ϑ tan 2815 (1988). 1228–1251 45, u sin ϑ 0 ϑ v cos λ ϑ + 8(2020). 78 901,  λ) ∂λ ∂ϑ . 0 v  srpy.J. Astrophys. , PNAS ϑ 0 nu e.Earth Rev. Annu. , λ 0 u
| = = 0 )to 0) (0, 65 869, 1o 12 of 11 (u, (Elsevier, v [12] [13] [15] [14] Res. is )

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