Global climate modeling of Saturn's atmosphere. Part II: multi-annual high-resolution dynamical simulations
Aymeric Spiga, Sandrine Guerlet, Ehouarn Millour, Mikel Indurain, Yann Meurdesoif, Simon Cabanes, Thomas Dubos, Jérémy Leconte, Alexandre Boissinot, Sébastien Lebonnois, Mélody Sylvestre, Thierry Fouchet
GGlobal climate modeling of Saturn’s atmosphere.Part II: multi-annual high-resolution dynamical simulations
Aymeric Spiga ∗ , Sandrine Guerlet , Ehouarn Millour , Mikel Indurain , YannMeurdesoif , Simon Cabanes , Thomas Dubos , J´er´emy Leconte , AlexandreBoissinot , S´ebastien Lebonnois , M´elody Sylvestre , and Thierry Fouchet address: Campus Pierre et Marie Curie BC99, 4 place Jussieu 75005 Paris, France Institut Universitaire de France (IUF), address: 1 rue Descartes, 75005 Paris, France Laboratoire des Sciences du Climat et de l’Environnement (LSCE/IPSL), Commissariat `a l’´energie atomique etaux ´energies alternatives (CEA), Centre National de la Recherche Scientifique, Universit´e Paris-Saclay, address:Campus du CEA - Orme des Merisiers, Saclay, France Laboratoire d’Astrophysique de Bordeaux (LAB), Univ. Bordeaux, Centre National de la Recherche Scientifique(CNRS), address: B18N, all´ee Geoffroy Saint-Hilaire, 33615 Pessac, France School of Earth Sciences, University of Bristol, address: Wills Memorial Building, Queens Road, Bristol BS8 1RJ, UK Laboratoire d’´Etudes Spatiales et d’Instrumentation en Astrophysique (LESIA), Observatoire de Paris,Universit´e Paris Sciences et Lettres (PSL), Centre National de la Recherche Scientifique (CNRS), SorbonneUniversit´e, Univ. Paris Diderot, address: 5 place Jules Janssen, 92195 Meudon, France
Version: August 2, 2019 ∗ Corresponding author: [email protected] a r X i v : . [ a s t r o - ph . E P ] A ug ighlights • A new Global Climate Model for Saturn with radiative transfer • High-resolution numerical simulations on a duration of 15 Saturn years • Results on zonal jets, waves, eddies in Saturn’s troposphere
Abstract
The Cassini mission unveiled the intense and diverse activity in Saturn’s atmosphere: banded jets,waves, vortices, equatorial oscillations. To set the path towards a better understanding of thosephenomena, we performed high-resolution multi-annual numerical simulations of Saturn’s atmo-spheric dynamics. We built a new Global Climate Model [GCM] for Saturn, named the SaturnDYNAMICO GCM, by combining a radiative-seasonal model tailored for Saturn to a hydrodynam-ical solver based on an icosahedral grid suitable for massively-parallel architectures. The impact ofnumerical dissipation, and the conservation of angular momentum, are examined in the model be-fore a reference simulation employing the Saturn DYNAMICO GCM with a / ◦ latitude-longituderesolution is considered for analysis. Mid-latitude banded jets showing similarity with observationsare reproduced by our model. Those jets are accelerated and maintained by eddy momentum trans-fers to the mean flow, with the magnitude of momentum fluxes compliant with the observed values.The eddy activity is not regularly distributed with time, but appears as bursts; both barotropic andbaroclinic instabilities could play a role in the eddy activity. The steady-state latitude of occurrenceof jets is controlled by poleward migration during the spin-up of our model. At the equator, a weakly-superrotating tropospheric jet and vertically-stacked alternating stratospheric jets are obtained inour GCM simulations. The model produces Yanai (Rossby-gravity), Rossby and Kelvin waves atthe equator, as well as extratropical Rossby waves, and large-scale vortices in polar regions. Chal-lenges remain to reproduce Saturn’s powerful superrotating jet and hexagon-shaped circumpolar jetin the troposphere, and downward-propagating equatorial oscillation in the stratosphere. ontents A.1 Exploring the impact of dissipation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47A.2 Angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
Introduction
It has been decades since Saturn’s meteorological phenomena observed by Earth-based and spacetelescopes, and the pioneering Voyager missions, are challenging the fundamental knowledge ofgeophysical fluid mechanics (e.g., Ingersoll, 1990; Dowling, 1995). Yet, a mission as richly instru-mented as Cassini (Porco et al., 2005), offering from 2004 to 2017 an unprecedented spatial andseasonal coverage of Saturn’s weather layer, brought a new impulse to the studies of giant planets’atmospheric dynamics (e.g., review papers by Del Genio et al., 2009; Showman et al., 2018a).In Saturn’s troposphere, the Cassini measurements confirmed the banded structure of alternat-ing westward (retrograde) and eastward (prograde) jets, which features a 450 m s − super-rotatingequatorial jet (Porco et al., 2005; Garc´ıa-Melendo et al., 2010; Studwell et al., 2018). Further-more, the Cassini instruments assessed the remarkable stability of the enigmatic hexagonal jetin the northern polar region (Baines et al., 2009; S´anchez-Lavega et al., 2014; Antu˜nano et al.,2015; Fletcher et al., 2018), with exquisite details on the structure of the turbulent polar vortex(Sayanagi et al., 2017; Baines et al., 2018). They also offered a detailed record of mid-latitudeconvective storms (Dyudina et al., 2007; del R´ıo-Gaztelurrutia et al., 2012) and vortices (Vasavadaet al., 2006; Dyudina et al., 2008; Trammell et al., 2016; del R´ıo-Gaztelurrutia et al., 2018), in-cluding a chain of infrared bright spots named the “String of Pearls” (Sayanagi et al., 2014) andan exceptional coverage of Saturn’s latest Great White Spot (Fischer et al., 2011; S´anchez-Lavegaet al., 2011; Sayanagi et al., 2013). Cassini observations of Saturn’s cloud layer was also employedto demonstrate the high rate of conversion of energy from eddies to jets (Del Genio et al., 2007; DelGenio and Barbara, 2012), to detail the structure of vorticity (Read et al., 2009a), and to exploreJupiter’s and Saturn’s atmospheric energetic spectra across spatial scales (Galperin et al., 2014;Young and Read, 2017), confirming pre-Cassini theoretical studies about geostrophic turbulenceand the inverse energy cascade (Sukoriansky et al., 2002). All those observations strongly suggestthat large-scale tropospheric banded jets emerge from forcing by smaller-scale eddies and wavesarising from hydrodynamical instabilities.In Saturn’s stratosphere, not only the Cassini instruments led to key discoveries, but thelongevity of the mission permitted a seasonal monitoring of the unveiled phenomena. Cassini’shighlights in atmospheric science for the stratosphere include a spectacular stratospheric warmingassociated with the 2010 Great White Spot (Fletcher et al., 2011b, 2012; Fouchet et al., 2016), anequatorial oscillation of temperature in Saturn’s stratosphere (Fouchet et al., 2008; Guerlet et al.,2011; Li et al., 2011) with semi-annual periodicity (Orton et al., 2008; Guerlet et al., 2018), and aseasonal monitoring of the meridional distribution of Saturn’s stratospheric hydrocarbons (Guerletet al., 2009, 2010; Sinclair et al., 2013; Fletcher et al., 2015; Sylvestre et al., 2015; Guerlet et al.,2015b), hinting at a possible interhemispheric transport of chemical species. Cassini measurementseven enabled to link a disruption in the downward propagation of the equatorial oscillation to the2010 Great White Spot occurrence (Fletcher et al., 2017). Analogies can be drawn between Saturn’sand the Earth’s stratospheres (Dowling, 2008). Saturn’s equatorial oscillation is reminiscent ofEarth’s Quasi-Biennal Oscillation and Semi-Annual Oscillation (Andrews et al., 1987; Baldwinet al., 2001; Lott and Guez, 2013; Guerlet et al., 2018), driven by the propagation and breakingof Rossby, Kelvin and inertio-gravity waves. The interhemispheric transport of chemical species,which may affect the hydrocarbons distribution, might be analogous to the Earth’s Brewer-Dobson4irculation (Butchart, 2014).In this stimulating observational context, new modeling efforts are needed to broaden theknowledge of Saturn’s atmospheric dynamics by demonstrating the mechanisms underlying theabove-mentioned observed phenomena. A great deal of past modeling work focused on the processesresponsible for the banded tropospheric jets. A major difficulty with a giant planet is that the depthat which the atmosphere merges with the internal dynamo region and the strength at which theatmospheric circulations couple with magnetic disturbances have remained poorly constrained byobservations (Ingersoll, 1990; Liu et al., 2008) until gravity measurements were recently performedon board Juno and Cassini (Kaspi, 2013; Galanti and Kaspi, 2017; Kaspi et al., 2018; Galanti et al.,2019). Two distinct modeling approaches have been adopted to account for Saturn’s troposphericjets: “shallow-forcing” climate models [see next paragraph for references] account for processes inthe weather layer (baroclinic instability, moist convective storms), while “deep-forcing” dynamo-like models (Heimpel et al., 2005; Yano et al., 2005; Kaspi et al., 2009; Heimpel and G´omezP´erez, 2011; Gastine et al., 2014; Heimpel et al., 2016; Cabanes et al., 2017) resolve convectionthroughout gas giants’ molecular envelopes. Contrary to deep models, shallow climate models havehad difficulties reproducing gas giants’ equatorial super-rotating jets. This has been overcome byincluding either bottom drag and intrinsic heat fluxes to simulate deep interior phenomena (Lianand Showman, 2008; Schneider and Liu, 2009; Liu and Schneider, 2010), or latent heating by moistconvective storms (Lian and Showman, 2010), although the simulated equatorial jets are still abouttwice as less strong in simulations than in observations (e.g., Garc´ıa-Melendo et al., 2010). Thesituation for off-equatorial jets is reversed, with better agreement with observations obtained byshallow models compared to deep models, although the latter can be modified to obtain morerealistic results (Heimpel et al., 2005). The recent results from the Juno mission for Jupiter (Kaspiet al., 2018; Guillot et al., 2018) and the Cassini mission for Saturn (Galanti et al., 2019) showthat banded jets extend several thousand kilometers below the cloud layer, i.e. deeper than whatshallow models consider and shallower than what deep models consider, which probably indicatesthat shallow and deep models have both their virtues to represent part of the reality.Here we adopt the approach of “shallow-forcing” climate modeling to study Saturn. In the lastdecade, the traditional approach using idealized modeling (Cho and Polvani, 1996; Williams, 2003;Vasavada and Showman, 2005) – which still has great value to study how baroclinic and barotropicinstabilities shape Saturn’s jets (Li et al., 2006; Kaspi and Flierl, 2007; Showman, 2007), includ-ing its polar hexagonal jet (Rostami et al., 2017) and central vortex (O’Neill et al., 2015) – hasbeen complemented by the development of complete three-dimensional Global Climate Models(GCMs) for Saturn and giant planets (Dowling et al., 1998, 2006; Lian and Showman, 2010; Liuand Schneider, 2010; Young et al., 2019a,b). A GCM is obtained by coupling a hydrodynamicalsolver of the Navier-Stokes equations for the atmospheric fluid on the sphere (the GCM’s “dynam-ical core”) with realistic models for physical processes operating at unresolved scales: radiation,turbulent mixing, phase changes, chemistry (the GCM’s “physical packages”). Most of those ex-isting GCM studies for Saturn address the formation of tropospheric jets by angular momentumtransfer through eddies and waves, often with either a theoretical approach aiming to address giantplanets’ atmospheric dynamics (Schneider and Liu, 2009; Lian and Showman, 2010; Liu and Schnei-der, 2010, 2015) rather than a focused approach aiming to address Saturn specifically, or with alimited-domain approach using a latitudinal channel enclosing one specific jet to explain structures5uch as the Ribbon wave or the String of Pearls (Sayanagi et al., 2010, 2014), to investigate theimpact of convective outbursts (Sayanagi and Showman, 2007; Garc´ıa-Melendo et al., 2013), or todiscuss the polar hexagonal jet (Morales-Juber´ıas et al., 2011, 2015). The idealized GCM ap-proach can also be employed to study equatorial oscillations in gas giants (Showman et al., 2018b).All those existing studies use simple radiative forcing rather than computing a realistic “physicalpackage” that includes seasonally-varying radiative transfer. The latter approach has been ex-plored to study Saturn’s stratosphere, either to constrain large-scale advection / eddy mixing inphotochemical models (Friedson and Moses, 2012), or to build a modeling framework applicableto extrasolar hot gas giants (Medvedev et al., 2013). Those studies of Saturn’s stratosphere makeuse, however, of prescribed, ad hoc , tropospheric jets.The existing body of work on “shallow-forcing” modeling has thus paved the path towards acomplete three-dimensional Global Climate Model (GCM) for giant planets. However, such a com-plete troposphere-to-stratosphere GCM for Saturn, capable to address the theoretical challengesopened by observations is yet to emerge. We propose that four challenges shall be overcome todevelop a complete state-of-the-art GCM for Saturn and gas giants. C The radiative transfer computations necessary to predict the evolution of atmospheric tem-perature, especially in the stratosphere, must be optimized for integration over decade-longgiant planets’ years, while still keeping robustness against observations. C Large-scale jets and vortices emerge from smaller-scale hydrodynamical eddies, through aninverse energy cascade driven by geostrophic turbulence. Relevant interaction scales (e.g.Rossby deformation radius) are only 1 ◦ latitude-longitude in gas giants vs. 20 ◦ on the Earth,making eddy-resolving global simulations over a full year four orders of magnitude morecomputationally expensive in gas giants. C Terrestrial experience shows that models need to extend from the troposphere to the strato-sphere with sufficient vertical resolution to resolve the vertical propagation of waves respon-sible for large-scale structures in both parts of the atmosphere. Moreover, a specific require-ment of giant planets is to extend the model high enough in the stratosphere to model thephotochemistry of key hydrocarbons impacting stratospheric temperatures (Hue et al., 2016). C Climate models cannot extend neither deep enough to predict how tropospheric jets interactwith interior convective fluxes and planetary magnetic field (Kaspi et al., 2009; Heimpel andG´omez P´erez, 2011), nor high enough to capture the coupling of stratospheric circulationswith thermospheric and ionospheric processes (M¨uller-Wodarg et al., 2012; Koskinen et al.,2015). A suitable approach to couple the weather layer with either the slowly-evolvingconvective interior, or the rapidly-evolving ionized external layers, remains elusive.Here we report the development and preliminary dynamical simulations of a new Saturn GCMat Laboratoire de M´et´eorologie Dynamique (LMD), which aims at understanding the seasonalvariability, large-scale circulations, and eddy & wave activity in Saturn’s troposphere and strato-sphere. It is a first step to further design a modeling platform dedicated to atmospheric circulationsof Saturn and other solar system’s giant planets. Challenge C about building fast and accurateradiative transfer for the Saturn GCM is addressed in Guerlet et al. (2014). In Guerlet et al.62014), which serves as Part I for the present study, a seasonal radiative–convective model of Sat-urn’s upper troposphere and stratosphere is described and the sensitivity to composition, aerosols,internal heat flux and ring shadowing is assessed, with comparisons to the observed thermal struc-ture by Cassini and ground-based telescopes. In this Part II paper, we address Challenge C byperforming high-resolution dynamical simulations with our Saturn GCM. Our GCM is built bycoupling the physical packages (notably, radiative transfer) of Guerlet et al. (2014) with DYNA-MICO, a new dynamical core developed at LMD which uses an original icosahedral mapping of theplanetary sphere to ensure excellent conservation and scalability properties in massively parallelresources (Dubos et al., 2015). We describe here the insights gained from GCM simulations athigh horizontal resolutions (reference at 1 / ◦ latitude/longitude, and tests at 1 / ◦ and 1 / ◦ ) withtwo unprecedented characteristics at those horizontal resolutions: inclusion of realistic radiativetransfer and long integration times up to fifteen simulated Saturn years.The paper is organized as follows. Notations are defined in Table 1. In section 2, we providedetails on the characteristics of our Saturn DYNAMICO GCM, and the assumptions and settingsadopted for the simulations discussed in subsequent sections, with appendix A featuring a necessaryanalysis of the impact of horizontal dissipation and the conservation of angular momentum inour Saturn GCM. In section 3, we describe the results obtained with our reference 15-year-long1 / ◦ Saturn DYNAMICO GCM simulation, with an emphasis on the driving and evolution ofjets in section 4. In section 5, we summarize our conclusions and draw perspectives for futureimprovements of our Saturn DYNAMICO GCM needed to fully achieve challenges C , C and C ,as it comes to no surprise that the present study is only a preliminary path towards fulfillingarguably ambitious scientific goals. As is reminded in the introduction, a GCM consists in coupling a dynamical core interfaced withphysical packages (or parameterizations). Our project to develop a Saturn GCM started by thedevelopment of the latter: the physical packages used in our GCM are described in full detail inGuerlet et al. (2014). Our model’s radiative computations are based on a versatile correlated- k method, suitable for any planetary composition (Wordsworth et al., 2010; Charnay et al., 2013;Leconte et al., 2013) with k -coefficients derived from detailed line-by-line computations using theHITRAN spectroscopic database (Rothman et al., 2013). Radiative contributions include the threemain hydrocarbons (methane, ethane and acetylene), the broad H -H and H -He collision-inducedabsorption (Wordsworth, 2012), and tropospheric and stratospheric aerosols layers. Our radiativecomputations also feature ring shadowing (appendix A in Guerlet et al., 2014) and account forinternal heat flux independent with latitude (section 2 in Guerlet et al., 2014).The spectral discretization of the correlated- k model is optimized for Saturn, with a particularemphasis on accounting for absorption and emission bands of stratospheric methane CH (theprominent driver of Saturn’s stratospheric heating), and other hydrocarbons produced by its pho-todissociation (ethane C H and acetylene C H , the prominent drivers of Saturn’s stratospheric7 oordinates t Time s( λ, ϕ ) Longitude, Latitude ◦ E, ◦ N( x, y ) WE coordinate, SN coordinate (local frame) m z Altitude m L s Saturn’s heliocentric longitude (0 ◦ N spring) ◦ Meteorological variables p Pressure Pa T Temperature K u Zonal wind component (W → E) m s − v Meridional wind component (S → N) m s − w Vertical wind component m s − m Atmospheric mass kg M Axial Angular Momentum (AAM) kg m s − µ Specific AAM (per unit mass, equation 18) m s − ζ Relative vorticity (vertical component ∂v/∂x − ∂u/∂y ) m s − q Ertel potential vorticity kg m s − K − Planetary parameters and model settings
Ω Rotation rate (cid:63) . × − s − ω Obliquity 26 . ◦ g Acceleration of gravity 10 .
44 m s − a Planetary radius 6 . × m c p Specific heat capacity 11500 J kg − K − M Molecular mass 2 .
34 g mol − R Ideal gas constant normalized with M . c p Φ i Internal heat flux 2 . − τ R Timescale for bottom Rayleigh drag 100 Earth days ϕ R Minimum latitude ( ± ϕ R ) for bottom drag 33 ◦ Computations f = 2 Ω sin ϕ Coriolis parameter at latitude ϕ s − β = ∂f∂y = ϕa Beta parameter (meridional derivative of f ) m − s − H = R Tg
Atmospheric scale height m N = gθ ∂θ∂z Brunt-V¨ais¨al¨a frequency s − ψ Axisymmetric component of variable ψ Zonal average ψ (cid:48) = ψ − ψ Eddy † component of variable ψ Zonal anomalyTable 1:
Physical quantities used in the paper. The numerical values provided in the right columncorresponds to the values set in our Global Climate Model (section 2.2). (cid:63)
The rotation rate corresponds to Saturn “days” of s, according to the value obtained inRead et al. (2009b) using an approach based on potential vorticity (denoted System IIIw). † Eddies are defined as deviations (perturbations) from the zonal-mean flow and can represent theeffects of turbulence, waves, and instabilities. − (Rey et al., 2018, in lieu of the Karkoschka and Tomasko (2010) band model) and thetwo main isotopes CH and CH D are now included. This improves the predicted temperaturesin the middle stratosphere (1 −
10 mbar) by about 2 K. The vertical profiles of hydrocarbons’abundances are held constant with latitude and season, and set as is described in Guerlet et al.(2014) using a combination of Cassini observations (Guerlet et al., 2009) and photochemical mod-eling (Moses et al., 2000). Variations up to 100% of acetylene abundance are observed at highlatitudes (Fletcher et al., 2015, note that Sylvestre et al. (2015) found weaker variations) whichwould entail temperature variations of a couple K in the vicinity of the 1-mbar level (Guerletet al., 2014, section 4.4); coupling our radiative model with a seasonal photochemical scheme isconsidered as a future development for dedicated middle-to-upper stratosphere GCM simulations(see Challenge C , as well as Hue et al., 2016, 2018).Guerlet et al. (2014) showed that this seasonal model allowed for both efficiency and accuracy,with satisfactory comparisons with Cassini measurements – including the observed temperature“knee” caused by heating at the top of the tropospheric aerosol layer, and the meridional gradientbetween the summer and winter stratosphere (Fletcher et al., 2010a, 2015). Temperatures predictedwith our Saturn DYNAMICO GCM are compared with Cassini measurements in section 3.1.The need to address specifically Challenge C (i.e. to achieve fine-enough horizontal resolutionsin order to predict the arising of smaller-scale eddies and the inverse cascade in the context ofgeostrophic turbulence) requires the use of a suitable dynamical core in our Saturn GCM. To thatend, we chose to employ DYNAMICO, which is developed at LMD as the next state-of-the-artdynamical core for Earth and planetary climate studies (Dubos et al., 2015), and tailored formassively parallel High-Performance Computing resources (scalability tested up to 10 cores).Our dynamical core DYNAMICO solves the primitive hydrostatic equations assuming a shallowatmosphere, i.e. z (cid:28) a (relaxing this assumption to solve the quasi-hydrostatic deep-atmosphereequations is considered for future developments of the model, Tort et al., 2015). The globalhorizontal mesh in DYNAMICO is set as a quasi-uniform icosahedral C-grid (Dubos et al., 2015)obtained by subdivision of a regular icosahedron: the total number of hexagonal cells is 10 × N × N corresponding to N × N sub-triangles subdividing each of the 10 main triangles of the icosahedrongrid ( N is the parameter by which the horizontal resolution is set in DYNAMICO). Control volumesfor mass, tracers and entropy/potential temperature are the hexagonal cells of the Voronoi meshto avoid the fast numerical modes of the triangular C-grid. Vertical coordinates are mass-basedcoordinates: “sigma” levels defined as p/p b where p b is the pressure at the bottom of the model.Spatial discretizations in DYNAMICO are formulated following an energy-conserving three-dimensional Hamiltonian approach (Dubos et al., 2015). Time integration is done by an explicitRunge-Kutta scheme (chosen for stability and accuracy). Subgrid-scale (unresolved) dissipationin the horizontal dimension is included as an additional hyperdiffusion term in the vorticity, di-vergence and temperature equations (see section A.1). In the vertical dimension, subgrid-scaledissipation is handled in the physical packages through a combination of a Mellor and Yamada(1982) diffusion scheme for small-scale turbulence, and a dry convective adjustment scheme fororganized turbulence (convective plumes, see section 2.4 of Hourdin et al., 1993). In the case ofour Saturn DYNAMICO GCM simulations, the adjustment scheme is the dominant term enablinga neutral profile in the troposphere. This simple adjustment scheme computes the temperature9endencies required to reach the entropy-conserving mixed layer of any convectively-unstable layerappearing in the model. Those characteristics entail that this scheme is not a source of small-scaleeddies in the model, which was checked in practice in our Saturn DYNAMICO GCM.The XIOS library (XML Input/Output Server, Meurdesoif, 2012, 2013) is employed to handleany input/output operations independently from the timeframe imposed by the numerical inte-grations: not only this improves the efficiency of the numerical integrations in massively-parallelcomputing clusters, but this also enables for complex operations on computed fields to be car-ried out during model runtime rather than as a post-processing operation. Notably, mapping thedynamical fields computed in the non-conformal icosahedral DYNAMICO grid towards a regularlatitude-longitude grid, using finite-volume weighting functions, is performed by XIOS directlyduring our GCM runtime. The simulations discussed in this paper are obtained from integrations with our Saturn DYNA-MICO GCM employing an horizontal icosahedral mesh with N = 160, corresponding to an ap-proximate horizontal resolution of 1 / ◦ in longitude/latitude (hereafter simply referred to as “1 / ◦ simulations”). Test simulations with N = 301 (1 / ◦ simulations) and N = 625 (1 / ◦ simulations),aimed at model testing rather than scientific exploration, are discussed in section 5 to open per-spectives for future work. The integration, dynamical timestep in the 1 / ◦ Saturn simulationsis 118 . / ◦ simulations feature 32 levels in the vertical dimension, ranging from p b ∼ C . This shallbe improved in further studies dedicated specifically to Saturn’s stratospheric phenomena (notably,the equatorial oscillation). Our DYNAMICO model features an optional absorbing (“sponge”)layer with a Rayleigh drag acting on the topmost model layers as a surrogate for gravity wave dragin the stratosphere, but we do not use it for the simulations presented in this paper, similarly toprevious studies (Schneider and Liu, 2009; Liu and Schneider, 2010). Indeed, Shaw and Shepherd(2007) showed that the inclusion of sponge-layer parameterizations that do not conserve angular10omentum (which is the case for Rayleigh drag), or allow for momentum to escape to space,implies a sensitivity of the dynamical results (especially zonal wind speed) to the choice for modeltop or drag characteristic timescale, because of spurious downward influence when momentumconservation is violated.Our bottom condition at the 3-bar level is similar to Liu and Schneider (2010). We includea simple Rayleigh-like drag d u/ d t = − u/τ R , with a timescale τ R = 100 Earth days. This dragplays the role devoted to surface friction on terrestrial planets, which allows to close the angularmomentum budget through downward control (Haynes and McIntyre, 1987; Haynes et al., 1991).This could also be regarded as a zeroth-order parameterization for Magneto-HydroDynamic (MHD)drag as a result of Lorenz forces acting on jet streams putatively extending to the depths of Saturn’sinterior (Liu et al., 2008; Galanti et al., 2019), much deeper than the shallow GCM’s model bottom.Whether or not including a bottom drag at 3 bars is physically justified is out of the scope of thepresent paper, and improving on this admittedly simplistic bottom boundary condition is an entireresearch goal on its own (part of what we named Challenge C ). In the present study, we take thisbottom drag as an imperfect, yet unambiguous, means to close the angular momentum budgetand accounting for deep-seated phenomena in shallow-forcing models for gas giants (Schneider andLiu, 2009; Liu and Schneider, 2010, 2015).Similarly to Liu and Schneider (2010), the bottom drag is not exerted at equatorial latitudes (i.e. | ϕ | < ϕ R = 33 ◦ ) as it artificially suppresses the cylindrical barotropic circulation structures thatdevelop along the rotational axis (Taylor columns). This approach mimics the so-called tangentcylinder, which is thought to cause the equatorial super-rotating jets in deep-convective models(Heimpel et al., 2005; Kaspi et al., 2009). The value ϕ R = 33 ◦ is obtained by the geometricalconstraint r R = a cos ϕ R , with r R = 0 . a (cid:39) (cid:39) r R isalso consistent with the 8000 − `a la Guerlet et al., 2014) using theexact same physical parameterizations and vertical discretization than the full Saturn DYNAMICOGCM integrations. The single-column model is initialized with an isothermal profile and run fortwo Saturn decades to ensure that the annual-mean steady-state radiative-convective equilibrium isreached, especially at the deepest layers at 3 bars (a couple Saturn years is usually enough to reachequilibrium in the stratosphere where radiative timescales are shorter than in the troposphere, seeFigure 1 and section 3.1). The initial zonal and meridional wind fields in our reference 1 / ◦ SaturnDYNAMICO GCM simulations are set to zero.Discussions on the impact of numerical dissipation and on the conservation of angular momen-tum in our DYNAMICO-Saturn model are respectively detailed in appendices A.1 and A.2.11
Atmospheric dynamics in our reference Saturn GCMsimulation
Hereafter are discussed the results of 15 complete Saturn years simulated by our Saturn DYNA-MICO GCM with 1 / ◦ longitude/latitude resolution. The analysis of angular momentum in appendix A.2 shows that the 15-year duration of simulationensures dynamical spin-up and that the dynamical fields are in quasi-steady state. Full radiativespin-up must also be ensured, along with dynamical spin-up. In Figure 1 the evolution of the meantemperature in the northern hemisphere is shown: as is expected from differences in radiativetimescales, the troposphere takes longer to reach steady-state seasonal cycle (about eight Saturnyears) than the tropopause level does (about three Saturn years). This shows that satisfactoryspin-up, both dynamical and radiative, is ensured starting from the ninth simulated year.Figure 1:
Evolution over the whole 15-year duration of the reference Saturn GCM simulationof zonal-mean temperature T , averaged over latitudes ϕ = 20 − ◦ N, in the tropopause region( p = 100 mbar, left) and in the troposphere ( p = 1 bar, right). The comparison of our seasonal radiative-convective model with observations, both from in-struments on board the Cassini spacecraft and ground-based telescopes, is discussed at length inGuerlet et al. (2014). Yet a sanity check is necessary, given that we now use this model interactivelywith a three-dimensional dynamical core.Figure 2 shows meridional-vertical sections of zonal-mean temperatures, both simulated byour Saturn DYNAMICO GCM and observed by the Cassini / Composite InfraRed Spectrometer(CIRS) instrument in 2015. The model successfully reproduces the vertical transition from tro-posphere to stratosphere, and the rather flat meridional gradients of temperature at this season( L s ∼ ◦ ). In Figure 3 (top), the simulated meridional-vertical section of the Brunt-V¨ais¨al¨a fre-quency N indicates that the radiative-convective transition, between the neutral profile ( N ∼ N >
0) in the upper troposphere and lower12igure 2:
Latitude-pressure section of zonal-mean temperature T (top) observed by Cassini/CIRSin 2015 (Guerlet et al., 2015a) vs. (bottom) produced by our Saturn DYNAMICO GCM in the four-teenth simulated Saturn year at L s ∼ ◦ (see Table 1 for a definition of L s ). The Cassini/CIRSobservations shown here are nadir retrievals, with optimal sensitivity in the − mbarand − . mbar ranges. Model results are similar should any of the simulated Saturn year startingfrom year eight be considered. −
600 mbar, which is in agreement with observations (P´erez-Hoyosand S´anchez-Lavega, 2006; Fletcher et al., 2007).Figure 3: (Top) Latitude-pressure section of zonal-mean Brunt-V¨ais¨al¨a frequency N and (bottom)meridional profile of zonal-mean temperature T at 500 mbar, averaged over the fourteenth simulatedSaturn year with our Saturn DYNAMICO GCM. Nevertheless, while a 1 − −
10 Ktoo warm compared to CIRS observations was also noted with one-column radiative-convective14odeling (Guerlet et al., 2014; Sylvestre et al., 2015) and is not a feature introduced by ourdynamical simulations. Putative dynamical effects (e.g. Brewer-Dobson seasonal circulations)had been proposed to explain this discrepancy between radiative models and CIRS observations;however, the adopted setting for our Saturn DYNAMICO GCM simulations does not allow us toaddress this question that would require to raise the model top above the 1-mbar level.The zonal jets produced by our dynamical model (discussed at length in what follows) are asso-ciated with distinctive temperature signatures, i.e. localized meridional gradients of temperature(see Figure 2 and 4), according to the thermal wind equilibrium which links ∂u/∂z to ∂T /∂y .These thermal signatures associated with jets are of similar amplitude between modeling and ob-servations, although the localization (i.e. latitude) of those thermal signatures is not compliantbetween models and observations, echoing the discrepancies in latitude between the observed andmodeled jet structures (see section 3.2). Figure 5 is a snapshot of the steady-state zonal flow of our 1 / ◦ Saturn DYNAMICO GCMsimulation. Our model produces mid-latitude zonal jets, both eastward and westward (i.e. progradeand retrograde), which average intensities over a year reach about 45 −
50 m s − for westward jetsand 60 −
65 m s − for eastward jets at the visible cloud deck (0 . − . − q θ calculated on isentropic surfaces – a conserved quantity(i.e. a flow “tracer”) for adiabatic motions (Vallis, 2006) – is defined under hydrostatic approxi-mation following Read et al. (2009a) equation 3 q θ = − g ( f + ζ θ ) ∂θ∂p (1)where f and ζ θ are defined as in Table 1, with the θ subscript denoting evaluation across a surfaceof uniform potential temperature θ . The meridional profile of PV associated with the troposphericjet structure simulated by our Saturn DYNAMICO GCM is shown in Figure 6. Characteristic PV“staircases” (i.e. sharp PV gradients) are found within the core of each mid-latitude eastward jet,surrounded by areas with “mixed PV” (i.e. uniform PV with latitude) on the flanks of the jets.The Ertel PV field obtained in Figure 6 with our model is similar to the one obtained through15igure 4: Meridional profiles of zonal-mean temperature T at the two opposite solstices explored bythe Cassini spacecraft (blue: year 2005, L s = 300 ◦ ; red: year 2015, L s = 70 ◦ ). Squares correspondto Cassini / CIRS limb retrievals (Guerlet et al., 2009, 2015a), crosses correspond to Cassini /CIRS nadir retrievals (Fletcher et al., 2007; Guerlet et al., 2015a), lines correspond to Saturn DY-NAMICO GCM simulations (fourteenth simulated year). Middle stratospheric conditions (2 mbar)are shown in the top plot, where both CIRS limb and nadir retrievals are valid; upper troposphericconditions (200 mbar) are shown in the bottom plot, where CIRS nadir retrievals are valid. Modelresults are similar should any of the simulated Saturn year starting from year eight be considered. Instantaneous zonal wind u in the beginning ( L s ∼ ◦ ) of the twelfth simulated year (after270 thousands simulated Saturn days), on the fifth sigma level of the model (pressure level ∼ . bar,corresponding to Saturn’s visible cloud deck and tropospheric conditions). Cassini measurements by Read et al. (2009a). This result, reminiscent of those obtained withidealized models of rapidly-rotating flows (Dunkerton and Scott, 2008; Dritschel and Scott, 2011;Marcus and Shetty, 2011), shows that the emergence and sharpening of mid-latitude eastward jetsis associated with PV mixing. This homogeneization of PV on the flanks of the jets is associatedwith the breaking of Rossby waves emitted at the core of the jet (e.g., Dritschel and McIntyre,2008). This creates a convergence of eastward momentum towards the regions of wave emission,i.e. the core of the eastward jets (Vallis, 2006; Schneider and Liu, 2009; Showman and Polvani,2011), helping to maintain the jet structure against dissipation (O’Gorman and Schneider, 2008).The vertical structure of the zonal-mean zonal jet system simulated by our Saturn DYNA-MICO GCM is displayed in Figure 7. The mid-latitude eastward and westward jets exhibit abarotropic structure (i.e. weak vertical shear) in the deep troposphere, and a baroclinic structure(i.e. significant vertical shear) in the upper troposphere / stratosphere. The latter is in balancewith the meridional temperature variations observed and modeled in the temperature structurein Figure 4. Interestingly, eastward mid-latitude jets simulated in our Saturn DYNAMICO GCMdoes not weaken from the cloud level around 1 bar to the upper troposphere, as is observed (Garc´ıa-17igure 6:
Instantaneous zonal-mean Ertel potential vorticity (top) and zonal wind (bottom) pro-jected on a surface of equal potential temperature
K, corresponding to tropospheric conditions.The time adopted for this plot is similar to Figure 5. The
K value of potential temperaturecorresponds to upper tropospheric conditions, higher in altitude than Figure 5; lower troposphericconditions are not accessible through this diagnostic due to the difficulty of defining the Ertel po-tential vorticity on isentropes in neutral conditions ( N ∼ ). Our Python code used to calculatepotential vorticity contains excerpts from the code by Barlow (2017). Instantaneous latitude-pressure cross-section of zonal-mean zonal wind u (colors) andzonal-mean temperature T (contours) in the beginning ( L s ∼ ◦ ) of the fifteenth simulated year(after about 342 thousands simulated Saturn days). Model results are similar should any of thesimulated Saturn year starting from year eight be considered. Accounting for the preferential zonal wavenumber n = 6 (hexagonal) mode in the circumpolarjet structure on Saturn is still an open question (Morales-Juber´ıas et al., 2011, 2015), given the nar-row parameter space which allows for this mode to predominate over other modes (Barbosa Aguiaret al., 2010; Rostami et al., 2017). The polar jet in our Saturn DYNAMICO GCM simulation (Fig-ures 5 and 12) has a different morphology than the other mid-latitudes jets, exhibiting meanderingwith time, which cause it to undergo latitudinal deformation and temporal variability. However,the meandering of our simulated polar jet is intense and very variable with time, with neithera n = 6 nor any mode n predominance. This is clearly at odds with the observed stable slowly-moving hexagonal jet in Saturn’s northern polar regions (S´anchez-Lavega et al., 2014; Antu˜nanoet al., 2015). The polar jet’s zonal wind speed, latitudinal position and width are, however, keyfactors to account for Saturn’s northern polar hexagon (Morales-Juber´ıas et al., 2011). The polar19et simulated by our Saturn GCM is both too weak and too equatorward (as in Figure 5) to pos-sibly lead to a predominance of the n = 6 mode. Furthermore, the polar jet’s temporal evolutioninfluenced by poleward migration causes it to break under intensified meandering by barotropicand baroclinic instability (see section 4). This obviously prevents any high-latitude jet to settle asa stable, wavenumber-6 hexagon-shaped, structure. Either the baroclinicity in the polar regions isnot realistic enough in our Saturn DYNAMICO GCM (this influence of vertical shear is discussedin Morales-Juber´ıas et al., 2015); and/or the central polar vortex is insufficiently resolved in oursimulations, hence too weak to stabilize the hexagonal shape of the polar jet against meandering(the influence of the central polar vortex is discussed in Rostami et al., 2017). A prograde equatorial jet is produced in the troposphere by our 1 / ◦ Saturn GCM simulation. Itis, however, severely underestimated by one order of magnitude compared to the observed value byCassini ( ∼ −
400 m s − in System IIIw, Read et al., 2009b; Garc´ıa-Melendo et al., 2010). Thelocal super-rotation index s associated with this equatorial jet writes (e.g., Read and Lebonnois,2018) s = µ m + µ w µ mϕ =0 − a cos ϕ (Ω a cos ϕ + u )Ω a − s ∼ . ϕ = ± ◦ ), while the observed superrotating index is an orderof magnitude larger (Read and Lebonnois, 2018, their table 1) and the observed equatorial jetextends towards latitudes ± − ◦ .The prograde, eastward, equatorial jet in the Saturn DYNAMICO GCM arises from accelera-tion caused by convergence of eddy momentum towards the equator (see equation 13 in section 4).Figure 8 shows that, within the equatorial super-rotating jet, the eddy momentum transport u (cid:48) v (cid:48) is positive south of the equator and negative north of the equator, meaning that waves and eddiescause a convergence of eastward momentum at the equator. Yet this equatorial acceleration bywaves and eddies is probably underestimated by our Saturn DYNAMICO GCM, given the re-sulting modeled jet being ten times less strong than the observed equatorial jet (Garc´ıa-Melendoet al., 2010). This is consistent with the latitudinal profile of Ertel PV shown in Figure 6, wherePV mixing by Rossby waves in equatorial regions appears not sufficient to yield a truly PV-mixedarea, as is the case for mid-latitude jets.Based on the existing literature (e.g., Gierasch et al., 2000; Lian and Showman, 2010), apossible source of this underestimate of Saturn’s equatorial superrotation could be the lack ofa parameterization for moist convection in our model (see, e.g., the work on Jupiter by Zuchowskiet al., 2009; Young et al., 2019b). Our GCM results are actually in contrast with the simulationsby Liu and Schneider (2010) which did not include an additional (moist) convective source, apartfrom the combination of internal heat flux and convective adjustment. The fact that the convectiveadjustment scheme in Liu and Schneider (2010) has a non-zero relaxation time (cf. Schneider andLiu, 2009, appendix B, section d), while ours is instantaneously adjusting, might be an element ofexplanation for this discrepancy. Using a non-zero relaxation time might emulate the convectiveoverturning time of dry and moist convective structures, confirming the need to add, in our Saturn20igure 8: Latitude-pressure cross-section of zonal-mean eddy momentum transport u (cid:48) v (cid:48) (colors)and zonal wind u (contours) averaged over the whole 15-year duration of the reference simulation.This temporal averaging is carried out because, contrary to other jets, the equatorial jet does notmigrate with time and the equatorial eddies develop continuously rather than through “bursts” (seeFigure 13 in section 4). DYNAMICO GCM, a (dry and moist) convective parameterization more sophisticated than oursimple convective adjustment scheme (e.g. thermal plume modeling like Hourdin et al., 2002; Rioand Hourdin, 2008; Cola¨ıtis et al., 2013) that will better represent the local dynamics underlyingconvective mixing and the impact thereof on the generation of waves and eddies.While the strengths of mid-latitude jets increase with altitude in our Saturn DYNAMICOGCM simulations (see section 3.2.1), the intensity of the simulated equatorial jet decreases withaltitude, which is also in line with the Cassini observations reported in Studwell et al. (2018).Although a quantitative comparison with observations is prevented by the severely underestimatedequatorial wind speed in our GCM, the fact that the equatorial jet decays from the cloud levelto the tropopause is also observed by Flasar et al. (2005), Li et al. (2008), and S´anchez-Lavegaet al. (2016). Figure 8 indicates that this decay is caused in our Saturn DYNAMICO GCMby divergence of eastward eddy momentum at the tropopause, in contrast to the convergenceof eastward eddy momentum causing the super-rotating jet in the troposphere. Interestingly,to interpret the Cassini VIMS observations of tracers, Fletcher et al. (2011a) proposed that twostacked, reversed cells are present in Saturn’s troposphere, one resulting from “jet damping” inthe upper troposphere and one resulting from “jet pumping” in the mid-troposphere (their section6). This is compliant with Figure 8 showing in equatorial regions an area of eddy divergence /westward jet sitting on top of an area of eddy convergence / eastward jet.Saturn’s equatorial stratosphere exhibits a downward propagation of (supposedly zonally-symmetric) alternating positive and negative temperature perturbations with respect to the radiative-21igure 9:
Same as Figure 7 for northern summer solstice ( L s ∼ ◦ , top panel) and fall equinox( L s ∼ ◦ , bottom panel). − ◦ → ◦ N, surrounded aboveand below by westward jets. This structure is reminiscent of the stacked jet signature derivedthrough thermal wind balance from Cassini observations of equatorial oscillation of temperature(e.g., Guerlet et al., 2018). Yet, contrary to Cassini observations, no downward propagation of thisjet signature is reproduced by our Saturn DYNAMICO GCM between northern summer and fall(Figure 9).Section 3.3.1 features a discussion of the equatorial waves being most probably responsiblefor the stacked jet structure in the stratosphere, and possible explanations for the lack of ver-tical propagation of the structure. A key point, not related to the wave analysis, that shall beemphasized here is that the model top is too low (at 1 mbar), and the vertical resolution of themodel in the stratosphere is too coarse, to correctly study the observed equatorial oscillations.For instance, the equatorial westward jet above 10 mbar is compliant with the thermal wind fieldderived by Fouchet et al. (2008), but our too-low model top prevents us from discussing the bulkof the observed oscillation above the 10-mbar pressure level.
A close examination of the tropical structure of zonal wind in Figure 5 hints at planetary waveactivity – notably a prominent wavenumber-2 signal. This is confirmed by Figure 10 in which theeddy (non-axisymmetric) components T (cid:48) , u (cid:48) , v (cid:48) are shown in the equatorial region at the tropopauselevel. The prominent wavenumber-2 signal features zonal wind and temperature perturbations(about 0 . λ / time t space to the zonal wavenumber s Instantaneous view on the zonal perturbations of temperature T (cid:48) (shaded contours) andhorizontal winds [ u (cid:48) , v (cid:48) ] (wind vectors) in the beginning ( L s ∼ ◦ ) of the twelfth simulated year (after270 thousands simulated Saturn days), on the fifteenth sigma level of the model (pressure level ∼ mbar, corresponding to Saturn’s tropopause, transition between troposphere and stratosphere).Only latitudes below ◦ N/S are shown (tropical channel). A similar figure is obtained at Saturn’scloud level ( ∼ . bar, as in Figure 5), with an even stronger prominence of the wavenumber-2signal. / frequency σ space, of the symmetric ( T S ) and antisymmetric ( T A ) components of the temperaturefield about the equator T S = 12 ( T ◦ N + T ◦ S ) T A = 12 ( T ◦ N − T ◦ S ) (3)(similar computations are also performed for zonal wind u and meridional wind v ). Our code usesthe Fast Fourier Transform package included in the scipy Python library. We validated indepen-dently our spectral analysis code on well-defined (semi-)diurnal tides and Kelvin waves simulatedin the Martian atmosphere (Wilson and Hamilton, 1996; Lewis and Barker, 2005; Guzewich et al.,2016).We perform the Fourier analysis on a specific 1000-day-long 1 / ◦ Saturn DYNAMICO GCMrun with frequent (daily) output, restarted from the GCM state after 270 thousands simulatedSaturn days (about 11 simulated Saturn years). The spectral mapping in the ( s, σ ) space enablesto evidence Rossby and Kelvin waves in the symmetric component T S and Yanai waves in theantisymmetric component T A (the former waves can also be detected in u S and the latter wavesin u A and v S , see Wheeler and Kiladis, 1999; Kiladis et al., 2009). Results for the temperaturefield simulated at the tropopause are shown in Figure 11, along with the dispersion relation forequatorial waves (Maury and Lott, 2014) √ γ (2 ν + 1) = γ σ − s − s/σ γ = 4 a Ω g h (4)24igure 11: Spectral analysis of the equatorial waves produced in our Saturn DYNAMICO GCM(in dynamical steady-state) at the same vertical level as Figure fig:eqwave. This spectral powermapping is obtained by a two-dimensional Fourier transform from the longitude λ / time t spaceto the zonal wavenumber s / frequency σ space (Wheeler and Kiladis, 1999) of the symmetriccomponent of temperature T S about the equator (left, see equation 3) and antisymmetric compomentof temperature T A about the equator (right). We follow the common assumption of a positivesign convention for frequency σ , meaning that s > modes are eastward-propagating and s < modes are westward-propagating. The dispersion relation for equatorial waves (equation 4) issuperimposed for meridional mode number ν = − , +1 (left panel, Rossby waves on the s < side and Kelvin waves on the s > side) and meridional mode number ν = 0 (right panel, Yanaiwaves on the part of the curves increasing with s ). Four values of equivalent depths (equation 5)are included: h = 5 km (blue), h = 10 km (purple), h = 20 km (magenta), h = 50 km (red). ν is the meridional mode number and defines the considered wave (Rossby: ν = +1 , +2 , . . . Yanai: ν = 0, Kelvin: ν = − , − , . . . ), γ is named the Lamb parameter, and h is an equivalentdepth associated with the vertical wavenumber mm = N g h − H (5)Dominant modes in the symmetric and antisymmetric components of the temperature and windfields are detailed in Table 2.The spectral analysis shows that (consistently in the three analyzed fields) the prominentwavenumber-2 signal is a westward-propagating Yanai wave with a period 6 days (frequency 60 ◦ longitude per day). Our analysis also evidences, both in the temperature and zonal wind fields,westward-propagating Rossby waves with wavenumbers s = − s = −
6, exhibiting long periodsof hundreds Saturn days and frequencies of a couple degrees longitude per day (the wavenumber- s = − s = +2 , +3 , +4, periods 10 −
20 Saturn days, and frequencies acouple tens of degrees longitude per day; the Kelvin wave signal is much fainter in the zonal windcomponent. This Kelvin wave signal is absent from the temperature field lower in the troposphere,at the cloud level.Elaborating from Voyager observations, Achterberg and Flasar (1996) detected a Rossby wave-number-2 signal in the tropics and mid-latitudes of Saturn’s tropopause (130 mbar), confinedby vertical variations of static stability. A similar signal is present in our our Saturn GCMDYNAMICO simulations: 1 degree longitude per day corresponds to ∼ . − , hence thesimulated Rossby wavenumber-2 signal has a phase speed of ∼ −
92 m s − , which is compatiblewith the phase speed of the order 100 m s − discussed by Achterberg and Flasar (1996). OurSaturn DYNAMICO GCM results indicate that other tropical Rossby modes (wavenumbers 3, 4,5, . . . ) are likely to be significant within Saturn’s tropics. This is compliant with the recent analysisby Guerlet et al. (2018), based on Cassini CIRS observations, which shows a complex structureat a pressure level of 150 mbar; interestingly, what appears as a wavenumber-3 Rossby modedominate in the upper stratosphere (0.5 to 5 mbar), possibly indicating conditions for breakingat this level or below for the other modes. Unless the wavenumber-2 signal found by Achterbergand Flasar (1996) is actually eastward-propagating at phase speeds about 1600 m s − (the fastplanetary modes were discarded by this study in favour of slower, more plausible, Rossby modes),the prominent wavenumber-2 westward-propagating Yanai wave in our Saturn DYNAMICO GCMsimulation remains to be evidenced in observations. A westward-propagating wavenumber-9 Yanaiwave mode has been, however, detected by Cassini CIRS in the upper stratosphere (1 mbar) by Liet al. (2008), but their observed temperature signature being symmetric about the equator (with amaximum at the equator) would be more compliant with a westward inertia-gravity wave (Guerletet al., 2018).The presence of equatorial vertically-propagating eastward Kelvin waves and westward Rossbyand Yanai waves in our Saturn DYNAMICO GCM simulations at the 130-mbar level means thatboth eastward and westward momentum is transferred in the stratosphere where vertically-stacked26ominant modes in T S s σ ( ◦ /d) period (d) log(SP)+2 22.0 16.3 7.9+3 32.1 11.2 7.6-3 3.5 102.0 7.5-6 8.2 43.8 7.4-4 5.0 72.4 7.3-2 1.4 262.7 7.3-5 6.8 53.2 7.2+4 42.5 8.5 7.2Dominant modes in u S s σ ( ◦ /d) period (d) log(SP)-6 8.2 43.8 9.8-2 3.5 102.0 9.7-3 3.2 113.6 9.6-4 4.3 84.7 9.6-5 5.3 67.5 9.4-7 8.9 40.3 9.4+2 22.0 16.3 9.3Dominant modes in T A , u A , v S s σ ( ◦ /d) period (d) log(SP)-2 59.0 6.1 10.1Table 2: Spectral modes detected by Fourier analysis and depicted in Figure 11. The spectralmapping for wind components is not shown, as it is similar to the spectral mapping for temperatureshown in Figure 11. SP stands for spectral power, d for Saturn days. The frequencies and periodsare intrinsic, i.e. with respect to a frame fixed on the zonally-averaged zonal flow u ∼ − m/s atlatitudes ◦ N/S. ± ◦ associated with thestacked jets ( ∼ ± −
10 K, figure not shown) are much lower than the contrasts obtained by Cassinithermal infrared measurements ( ± −
20 K, Fouchet et al., 2008; Guerlet et al., 2018). Thosediscrepancies with observations are probably related to a weak transfer of momentum to the meanflow by the resolved waves in our model:( a ) while our spectral analysis reveals a Kelvin-wave signal at the tropopause, moist convection inthe deep troposphere of Saturn, not accounted for in the current version of our model, couldcause convectively-coupled Kelvin Waves, which are an important component to explain theQuasi-Biennal Oscillation on Earth (Kiladis et al., 2009);( b ) mesoscale (inertia-)gravity waves are not resolved by our model and are known to contributeto the momentum flux responsible for equatorial oscillations on Earth (Lindzen and Holton,1968; Lott and Guez, 2013; Maury and Lott, 2014) and this possibility has also been exploredin Jupiter’s stratosphere (Cosentino et al., 2017);( c ) the absence of a strong equatorial super-rotating jet in our simulations means that the verticalpropagation (and possible filtering) of equatorial waves towards the stratosphere is differentbetween our simulations and the actual Saturn’s atmosphere.According to terrestrial modeling studies (e.g. Takahashi, 1996; Nissen et al., 2000; Watanabeet al., 2008; Lott et al., 2014), our modeling setting is ultimately lacking two key elements toreproduce Saturn’s equatorial oscillation: the model top must be raised to cover the stratosphericlevels (0 . −
10 mbar) where the stratospheric oscillation is observed, and the vertical resolutionmust be refined in the stratosphere. Those improvements, related to Challenge C , are deferredto a dedicated future study of Saturn’s stratospheric circulations using our Saturn DYNAMICOGCM (Bardet D. et al., Part IV in preparation). Our reference simulation with the Saturn DYNAMICO GCM exhibits a variety of extratropicaleddies, as is evidenced in Figure 12.The simulated tropospheric fields in Figure 12 indicate that the mid-latitude eastward jets atlatitude 30 ◦ N and 60 ◦ N are prone to meandering caused by high-wavenumber waves. Those wavesare found at the center of the eastward jets, featuring a strong inversion of the meridional gradientof potential vorticity (Rayleigh-Kuo necessary condition for barotropic instability, see section 4.4).A spectral analysis on a 2000-day sample of the temperature and wind fields, performed similarlyto the analysis in section 3.3.1 (except for a Doppler-shift correction considering the ambienteastward zonal jet u = 60 m s − ) indicates that the 30 ◦ N perturbations correspond to a westward-propagating wavenumber-19 Rossby wave with a period of 435 days (frequency 0 . ◦ longitude28igure 12: Instantaneous view on (top left panel) the zonal wind u and (top right panel) themeridional wind v on the fifth sigma level of the model (pressure level ∼ . bar), as in Figure 5and (bottom panel) relative Ertel potential vorticity (PV) on tropospheric isentrope θ = 205 Kafter 171 thousands simulated Saturn days (beginning of the seventh simulated year). Ertel PVcomputations are described in equation 1 and section 3.2.1. − ). The characteristics of this wave (wavenumber, phase speed, andoccurrence at the center of a mid-latitude eastward jet) are very similar to the idealized modelingresults obtained by Sayanagi et al. (2010) to explain the “Ribbon wave”, a Rossby wave propagatingin the extratropical latitudes of Saturn as a result of barotropic and baroclinic instability (Godfreyand Moore, 1986; Sanchez-Lavega, 2002; Gunnarson et al., 2018). Furthermore, the meanderingphase speed and wavenumber reproduced by our Saturn DYNAMICO GCM in the 30 ◦ N eastwardjet are compliant with the slow ribbon waves identified by Gunnarson et al. (2018) with Cassiniimaging (their Figure 3d).It is worthy of notice that our Saturn DYNAMICO GCM simulations exhibit a chain of aboutten cyclonic vortices with an horizontal extent of a couple thousand kilometers (the successive redspots of positive vorticity in Figure 12 bottom) in the equatorward (southern) edge of the 30 ◦ Neastward jet. This signature shares the characteristics of the “String of Pearls” observed byCassini through infrared mapping (Sayanagi et al., 2014). Nevertheless, the simulated vorticesare more short-lived (typically a ten-day duration) than the observed cyclonic vortices, so theanalogy between the modeled structures and the “String of Pearls” is not complete.The most distinctive and large-scale vortices are found in polar regions in our Saturn DY-NAMICO GCM simulations. Figure 12 demonstrates a clear and sharp transition between themid-latitudes where the vorticity field is dominated by eddies, and the polar regions where thevorticity field is dominated by large-scale vortices. This is somewhat compliant with the pic-ture drawn by observations, where no vortex activity was observed in mid-latitudes before theappearance of the 2010 giant storm (Trammell et al., 2016). Both anticyclones and cyclones areproduced in our model: the excerpt shown in Figure 12 comprises four anticyclones and one largercyclone. Those simulated large-scale vortices exhibit a strong temporal variability, with mergingphenomenon combined with beta-drift effect (poleward for anti-cyclones, see Sayanagi et al., 2013),which causes their typical duration to be no more than several hundreds Saturn days. The picturedrawn by Figure 12 is typical of most of our Saturn DYNAMICO GCM simulation: anticyclonesappear favored against cyclones, which is in agreement with the putative longer stability of anti-cyclones compared to cyclones, but at odds with the statistics derived from Cassini imagery overseven Earth years by Trammell et al. (2016). A more in-depth analysis of the large-scale vorticesis out of the scope of this paper; furthermore, accounting for moist convection and the associatedrelease of latent heat appears to be a crucial addition to carry out this analysis (O’Neill et al.,2015).The last class of eddies are the remainder of non-axisymmetric disturbances that are neitherwaves nor vortices. Such “non-organized” eddies can be seen in Figure 12 between latitudes 30 ◦ Nand 60 ◦ N. Their activity can strongly vary with time: Figure 12 shows a case with a “burst” ofeddy activity at those latitudes, while the eddy activity can be close to none at other times in thesimulation (Figure 5). The presence of intermittent bursts of eddy activity is reminiscent of theresults of Panetta (1993) and is further discussed in section 4.1.30
Evolution of the tropospheric jet structure
The evolution of tropospheric jets with time in the 15-year duration of our reference 1 / ◦ SaturnDYNAMICO simulation is summarized in Figure 13. It takes about about 6-7 simulated Sat-urn years for the jet system to reach what most closely resembles a steady-state equilibrium; asimilar conclusion was drawn from the analysis of the temporal evolution of AAM (Figure 22 insection A.2). The zonal mean of the Eddy Kinetic Energy (EKE) e = 12 (cid:16) u (cid:48) + v (cid:48) (cid:17) (6)is also shown in Figure 13 to diagnose eddy activity.The first years of our Saturn DYNAMICO GCM simulations follow the evolution of zonal jetstypically obtained with nonlinear analytical models prone to barotropic and baroclinic instability.The evolution of our Saturn DYNAMICO GCM displayed in the top panel of Figure 14 is similarto the evolution described, e.g., in Figure 10 of Kaspi and Flierl (2007). The first simulated half-year exhibits no particular zonal organization (except at the equator). In the second half of thefirst simulated year, the growth of the fastest unstable mode leads to the emergence of numerousweak zonal jets, which subsequently reorganize, as additional growing eddy modes are present, tolead to a system with less, and wider, jets. The abrupt transition around 1.15 simulated yearsin Figure 13 is associated with significant eddy activity. The merging of numerous weak jetsinto a final jet structure with lesser and stronger jets is typical of the inverse energy cascade bygeostrophic turbulence which shapes the jet structure (Cabanes et al., 2017). Another typicalfeature of the inverse energy cascade shown in Figure 15 is the overall correlation between theRhines scale (Rhines, 1975) L β = 2 π (cid:115) Uβ with U = √ e (7)and the eastward jets’ width/spacing (Chemke and Kaspi, 2015b), with a tendency for broaderjets and increased spacing between jets towards higher latitudes (Kidston and Vallis, 2010). A fullexploration of the dynamical regimes (e.g., zonostrophy) and the inverse energy cascade in ourSaturn DYNAMICO GCM requires detailed spectral analysis of the flow energetics (Sukorianskyet al., 2002; Galperin et al., 2014; Young and Read, 2017) that will be developed in a follow-uppaper (Cabanes S. et al., Part III submitted).The most prominent feature of Figure 13 between the simulated years 1 and 6 is the polewardmigration of mid-latitude jets. This echoes the idealized simulations detailed in Chemke andKaspi (2015a) (see also jovian simulations by Williams, 2003). The migration is gradual, but thejet migration is much stronger in short-lived episodes characterized by a burst of eddy activity,which implies momentum transfer to the jet altering its meridional structure. A detailed analysisof a typical poleward migration episode in Figure 13 is proposed in section 4.3. The tropical jetsundergo a much weaker migration than the mid-latitude jets. Most of the migration eventsare poleward, but there is at least one clear equatorward migration event of a jet appearing at31igure 13: Evolution of the zonal-mean zonal wind u (top plot) and the zonal mean of the EddyKinetic Energy (EKE) / u (cid:48) + v (cid:48) ) (bottom plot) in Saturn troposphere (800 mbar) within thewhole 15-year duration of our Saturn DYNAMICO GCM simulation. Same as Figure 13 with an emphasis on (top) the first four and (bottom) the last foursimulated Saturn years of the full 15-year duration of our Saturn DYNAMICO GCM simulation.
Meridional profiles of the Rhines scale L β defined as in equation 7 (orange line)and eastward jet speed u (blue line). Those quantities are vertically-averaged in the pressurerange [10 , ] Pa. The left plot shows a temporal average within half of the second simulatedyear (during dynamical spin-up and strong jet migration) and the right plot within half of thetwelfth simulated year (quasi-steady state). The correlation between the Rhines scale and the jetwidth / spacing is more obvious in the former than in the latter. The left plot is computed inconditions of strong poleward migration of the eastward jets, a temporal evolution resembling thesimulations in Chemke and Kaspi (2015b). latitude 15 ◦ N in the beginning of year 7. This equatorward migration contributes to acceleratethe equatorial jet. This was also noticed by Young et al. (2019a) in their Jupiter GCM, althoughtheir simulations exhibit a global tendency for equatorward jet migration, contrary to the globaltendency for poleward jet migration found in our GCM simulations and in Chemke and Kaspi(2015a).The poleward migration of the mid-latitude jets continues until the migration rate slows downand the zonal jets reach their final latitude of occurrence in our Saturn DYNAMICO GCM simu-lation. Starting from the seventh simulated year to the fifteenth simulated year (Figures 13 topand 14 bottom), we notice a continuing, very slow poleward migration of the mid-latitude eastwardjets and equatorward migration of the tropical jets. The 15-year duration of our Saturn DYNA-MICO GCM simulation allows us to reach robust conclusions about the overall steady-state jets’structure and intensity; the long duration of our simulations also allows us to conclude that thisjet structure is still impacted by a slowly-evolving transient state over timescales of tens of Saturnyears. It is important to note here that we are not interpreting our Saturn GCM simulations asindicative of a current migration of Saturn zonal jets, which is not supported by observations.Our GCM simulations start with a zero-wind state which is not encountered in the actual Saturn’satmosphere, and may have never been encountered in the past history of Saturn’s atmosphere.Thus, we can only speculate that jet migration could have been an important process in the pastevolution of Saturn’s atmosphere and might explain the present latitudes of Saturn’s zonal jets.The bursts of eddy activities are also associated with acceleration of the zonal jets, would it34e a case of a migration episode or not (once the jets have migrated, the impact of the eddies isactually solely jet acceleration and no longer migration). This suggests that eddy forcing plays agreat role in shaping the zonal jets in Saturn’s atmosphere, as argued by existing modeling studies(Showman, 2007; Lian and Showman, 2010; Liu and Schneider, 2010) and Cassini observations(Del Genio et al., 2007; Del Genio and Barbara, 2012). We discuss this matter in section 4.2 for aglobal analysis and section 4.3 for a local analysis. Despite our Saturn DYNAMICO GCM resolvingthe seasonal evolution of Saturn’s troposphere and stratosphere, there is no clear seasonal trendassociated with the bursts of mid-latitude eddy activity (although the typical timescale betweenthe bursts is close to one year). This is in line with theoretical studies supporting abrupt stochastictransitions in the zonal jet structure prone to barotropic and baroclinic instabilities (Bouchet andSimonnet, 2009; Bouchet et al., 2013). The typical timescale between bursts is not so much set bythe seasonal cycle, but by the typical life cycle of instabilities (Panetta, 1993).
To investigate the mechanism by which zonal banded jets arise in our Saturn DYNAMICO GCMsimulation, we first consider a global diagnostic, the conversion rate C of eddy-to-mean kineticenergy. Cloud tracking with Cassini Imaging Science Subsystem (ISS) images has been employedto address the driving of Saturn’s zonal jets by eddy momentum fluxes (Del Genio et al., 2007; DelGenio and Barbara, 2012). The conversion rate C in m s − (or W kg − ), estimating the conversionper unit mass of eddy kinetic energy to zonal-mean kinetic energy, can be obtained by multiplyingeddy momentum transport u (cid:48) v (cid:48) by the meridional curvature of the zonal flow ∂u∂y . C = u (cid:48) v (cid:48) ∂u∂y (8)Wind observations by cloud tracking exhibit a globally positive conversion rate C >
0, both atthe middle troposphere ammonia cloud at 1 bar and the upper troposphere haze at 100 mbar,which suggests that Saturn’s zonal jets are eddy-driven (Del Genio and Barbara, 2012). Thefact that C is positive means that the eddy flux is, on average, equatorward in cyclonic shearregions and poleward in anticyclonic shear regions (Del Genio and Barbara, 2012), hence eastwardjets are accelerated by the convergence of eddy flux while westward jets are decelerated by thedivergence of eddy flux (see also PV discussions in section 3.2.1). The values of kinetic energyconversion rate observed by Cassini by Del Genio and Barbara (2012) (their Figure 11) are large: C ∼ × − W kg − at 100 mbar, and four times larger in the troposphere at 1 bar. Thoseestimates of the energy conversion rates support the idea that eddy momentum transfers are ableto maintain jets against dissipation. Similar conclusions were reached for Jupiter’s weather layerby Salyk et al. (2006).The positive conversion rates C simulated by our Saturn DYNAMICO GCM, shown in Fig-ure 16, indicate that our model supports the conclusion of Del Genio and Barbara (2012) thatSaturn’s zonal banded jets are, for a significant part, driven and maintained by eddies in theweather layer (see also Read et al., 2009a). This also confirms the diagnostics obtained from previ-ous GCM studies (Lian and Showman, 2008; Liu and Schneider, 2010). In the upper troposphere35igure 16: Rate per unit mass C in m s − (or W kg − ) estimating the conversion of eddy kineticenergy to zonal-mean kinetic energy. Vertical profiles of C are shown, anually-averaged (one lineper simulated year, from year 2 to 15) and globally-averaged ( ◦ S to ◦ N latitudes) This spatialand seasonal averaging is chosen to allow a direct comparison with Cassini estimates of C in DelGenio and Barbara (2012). haze layer at 100 hPa, our model predicts C ∼ . × − m s − , which matches to the order-of-magnitude the quantitative estimates obtained from Cassini by Del Genio and Barbara (2012),implying a typical timescale of replenishing the jets of less than a Saturn year. This shows thatour Saturn DYNAMICO GCM resolves a satisfactory conversion rate from eddies to zonal jets inthe tropopause, providing support for a “downward control” of jets at deeper levels (Haynes et al.,1991) by eddy forcing in the radiatively-driven upper troposphere, as proposed by Schneider andLiu (2009) and Liu and Schneider (2010). Nevertheless, our modeled values for C are half thoseobtained by cloud tracking on board Cassini, indicating room for improvement in predicting theeddy activity and jet curvature resolved by our GCM in the upper troposphere, suggesting theneed for either more accurate radiative computations, or an additional physical process causingeddies.The conversion rate C increases with altitude in our Saturn GCM simulation, whereas it de-creases with altitude in the Cassini observations of Del Genio and Barbara (2012). In other words,36f our simulations match the observations at 100 mbar, the conversion rate at the cloud layer is oneorder of magnitude lower in the Saturn GCM simulations than it is in the observations. Del Genioand Barbara (2012) already noticed this discrepancy by comparing their data to the GCM resultsof Liu and Schneider (2010). We speculate that our simulated eddy forcing of jets being compliantwith observations in the radiatively-driven tropopause, but not in the deeper troposphere, indi-cates that a source of tropospheric eddy forcing (e.g. latent heat release and convective motionsassociated with moist processes Zuchowski et al., 2009; Lian and Showman, 2010) is missing inour Saturn GCM. This is also consistent with our equatorial jet super-rotating too weakly for apossible lack of convectively-generated Rossby waves (Schneider and Liu, 2009). The approach using kinetic energy conversion rate in section 4.2 is to be understood on a globalsense; here we present diagnostics for eddy-induced jets that bear a more local sense.We can consider, as a typical and particularly illustrative example, the impact of the strongburst of eddy activity taking place in the northern hemisphere within 1000 Saturn days between 1 . . ∂u∂t = (cid:34) ∂u∂t (cid:35) R + (cid:34) ∂u∂t (cid:35)
E + X (9)where X is a mean nonconservative force such as e.g. diffusion, and the respectively “residual-mean” and “eddy-related” terms are written (e.g. equation 2.5 in Andrews et al., 1983) (cid:34) ∂u∂t (cid:35) R = − (cid:34) a cos ϕ ∂u cos ϕ∂ϕ − f (cid:35) v − ∂u∂p ω (10) (cid:34) ∂u∂t (cid:35) E = − a cos ϕ ∂u (cid:48) v (cid:48) cos ϕ∂ϕ − ∂u (cid:48) ω (cid:48) ∂p (11)The contributions of each of those terms, within the considered 1000 Saturn days prone to signif-icant eddy activity, are provided in Figure 17. As was also noticed by Lian and Showman (2008,their Figure 8), the two terms described in equations 10 and 11 are generally anti-correlated, whichindicates that a significant part of eddy-related acceleration (which might reach 2 × − m s − on average over 1000 Saturn days) contributes to maintaining an associated meridional circula-tion, in addition to contributing to the zonal jets. Under the assumption of steady-state zonaljets ( ∂u/∂t (cid:39) ◦ N eastward jet is slightly accelerated, in a situation where the residual-mean and eddy-related terms almost compensate; the 35 ◦ N eastward jet does not accelerate at its37igure 17:
Meridional profiles of the zonal-mean acceleration terms in the Eulerian-mean for-malism described by equation 9. The analysis is carried out after 41 thousands simulated Saturndays (around . simulated year), corresponding to a typical burst of eddies in the northern hemi-sphere evidenced in Figure 13. The plots are obtained after averaging over 1000 Saturn days. Theresidual-mean term (equation 10) is shown in the red line and the eddy-related term (equation 11)is shown in the green line. The net resulting acceleration is shown in the black full line and theactual acceleration obtained within the considered 1000 days is shown in the dashed blue line. Thecomparison of the latitudinal profiles of zonal-mean zonal winds at the beginning and in the end ofthe 1000-day sequence are shown respectively in light blue and orange in the bottom plot. ◦ Neastward jet undergoes strong eddy perturbations, especially on its poleward flank, not compen-sated by an evolution of the residual-mean circulation, which results in both a poleward migrationand an acceleration of this eastward jet. Note that, concomitantly with this overall accelerationof the eastward jets, westward jets are decelerating. This supports the interpretation proposed insections 3.2.1 and 4.2, and in the literature (e.g., Schneider and Liu, 2009), that there is a nettransfer of momentum from the westward jets towards the eastward jets.A complementary framework to study the evolution of jets – especially the eddy-related ac-celeration – is the Transformed Eulerian Mean approach. In this approach, the Eliassen-Palm(EP) flux F (e.g., Vallis, 2006), which meridional component writes in isobaric coordinates (e.g.equation 2.7 in Andrews et al., 1983) F ϕ = a cos ϕ (cid:32) − u (cid:48) v (cid:48) + ψ ∂u∂p (cid:33) with ψ = − v (cid:48) T (cid:48) / (cid:32) R Tc p p − ∂T∂p (cid:33) (12)provides a direct link between the convergence / divergence of eddy momentum and the resultingacceleration / deceleration of zonal jets. The horizontal contribution of eddies to zonal-mean windacceleration is the divergence of the meridional component F ϕ of the EP flux (cid:34) ∂u∂t (cid:35) eddies = 1 a cos ϕ ∂ F ϕ cos ϕ∂ϕ (cid:32) (cid:39) − a cos ϕ ∂u (cid:48) v (cid:48) cos ϕ∂ϕ (cid:33) (13)(The vertical contribution of eddies to zonal-mean wind acceleration is omitted in this equationbecause, in the specific context of our analysis of eddy-driven jets, it was found to be negligible).Weuse the expression in equation 12 to diagnose the eddy-driven acceleration in our Saturn GCMsimulations; we note, however, that the approximate expression in parenthesis (used e.g. to inter-pret Figure 8 in section 3.2.2) is reasonable in a vertically-integrated quasi-geostrophic framework,with zonal averaging making mean momentum flux convergence terms to be small compared tothe eddy momentum flux convergence terms (Hoskins et al., 1983; Vallis, 2006; Chemke and Kaspi,2015a).As was found from analyzing Figure 17, the 35 ◦ N eastward jet in the end of the first simulatedyear typically undergoes a poleward migration associated with a burst of eddy activity; this eddy-driven migration is continuing in the beginning of the second simulated year. Figure 18 indicatesthat the divergence of the Eliassen-Palm flux associated with this eddy activity indeed acts toslow down the jet core and accelerate its flanks, with a larger acceleration being experienced inthe poleward side.
The importance of barotropic and baroclinic instabilities has been discussed in the existing lit-erature as the source for gas giants’ banded jets (Dowling, 1995; Liu and Schneider, 2010) andthe evolution thereof, notably migration (Williams, 2003; Chemke and Kaspi, 2015a). Just as thevertical gradient of potential temperature enables to assess convective instability, the meridionalgradient of potential vorticity enables to assess barotropic / baroclinic instability (Dowling, 1995;39igure 18:
An episode of the poleward migration of a mid-latitude jet is shown on the top twofigures offering a magnified view of a relevant portion of Figure 13 showing an episode of jetmigration. The latitudinal profiles of the zonal-mean zonal jet at the beginning and in the endof the temporal interval spanned by the top plots are shown as middle plots (pressure level is mbar). The bottom plot features the acceleration term obtained by equation 13 (computationof the divergence of the Eliassen-Palm flux). (cid:34) ∂q∂y (cid:35) BT = β − ∂ u∂y (14)changes sign in the domain interior. The Charney-Stern-Pedlosky [CSP] necessary condition forbaroclinic instability is that the full-baroclinic meridional gradient of PV (cid:34) ∂q∂y (cid:35) BC = (cid:34) ∂q∂y (cid:35) BT − ρ ∂∂ Z (cid:34) ρ f N ∂u∂ Z (cid:35) Z = − H ln( p/p ) (15)either: changes sign in the interior [CSP1, similar to RK], is the opposite sign to u Z at the upperboundary [CSP2], is the same sign as u Z at the lower boundary [CSP3], or is zero and u Z isthe same sign at both boundaries [CSP4]. The CSP criterion is not defined in a neutral layer(where N ∼
0) such as Saturn’s troposphere, thus we carry out the analysis of the two necessaryconditions in equations 14 and 15 near the tropopause level in our simulations.The necessary condition RK for barotropic instability can be assessed from Figure 19 by deter-mining when the quantity described by equation 14 changes sign. In the first four simulated years,the mid-latitude eastward jets migrating from 30 ◦ N/S to 60 ◦ N/S fulfil the RK condition on thepoleward flanks, but not on the equatorward flank. In subsequent years (from year 5 to year 15)in our Saturn DYNAMICO GCM simulations, those eastward jets fulfil the RK on both flanks,which is also the case for the weakly-migrating 20 ◦ N/S jets throughout the 15-year simulation.The simulated mid-latitude jets in our Saturn DYNAMICO GCM are thus possibly impacted bybarotropic instability; this could be expected from the PV mapping in Figure 12 which clearlyshows that strong inversions of the meridional gradients of PV are found at the location of theeastward jets. This provides an explanation for the extratropical eddies found in the bulk of east-ward jets and discussed in section 3.3.2. It is worth reminding here that barotropic instability actsto transfer momentum from jets to eddies, and not the contrary. Hence the positive conversionrate C indicating an overall transfer from eddies to jets (section 4.2) hints at baroclinic instabilitycomplementing barotropic instability.Referring to the CSP1 criterion, Figure 19 shows that the conditions for baroclinic instabilityare met in polar regions – this is also true for barotropic instability according to the RK criterion.In mid-latitude eastward jets, the CSP1 necessary criterion for baroclinic instability is howeveronly verified in the poleward flank of the jets; the quantity described in equation 15 is mostlypositive at all latitudes (outside polar regions). The fact that the CSP1 criterion for baroclinicinstability is fulfilled in the poleward flank of the mid-latitude jets, and less so in the equatorwardflank, echoes the conclusions of Chemke and Kaspi (2015a) who demonstrated that the polewardmigration of jets in idealized GCM simulations of high-rotation planets is caused by a polewardbias in baroclinicity across the width of the jet.The condition CSP3 is fulfilled in the simulated mid-latitude eastward jets since the (baroclinic)meridional gradient of PV is of the same sign as the vertical shear u Z , i.e. positive as is shownin Figure 19 (see also section 3.2.1). Those simulated jets could thus be baroclinically unstable . CSP3 is often the decisive condition in the terrestrial environment too, see Vallis (2006). An assessment of
Evolution of the meridional gradients of zonal-mean potential vorticity (top left:barotropic, equation 14 for necessary condition RK, top right: baroclinic, equation 15 for necessarycondition CSP1) and vertical shear of zonal-mean zonal wind (bottom) at Saturn’s tropopause(100 mbar) within the whole 15-year duration of our Saturn DYNAMICO GCM simulation. Thediagnostic in equation 14 cannot be computed in the troposphere where the atmosphere is neutral( N ∼ ) and barotropic; the two other diagnostics are shown at the tropopause for consistency butare similar at deeper levels. C inFigure 16 and section 4.2, even if the order of magnitude compared to observations is correct. Forinstance, the fact that baroclinic instabilities cause unrealistically strong jet meandering in polarregions may be deemed an indication that meridional gradients could be improved at the bottomof our model. Deepening the model bottom and ensuring a more realistic baroclinicity in the lowertroposphere is an area of future improvement of our Saturn DYNAMICO GCM, as is the case formost existing shallow-atmosphere models. The conclusions of our study can be summarized as follows.1. The Cassini mission opened novel questions on tropospheric and stratospheric circulationson Saturn, with new modeling challenges (section 1 and challenges C C C C ).2. The Global Climate Model (GCM) we built is named the Saturn DYNAMICO GCM andcouples the radiative transfer of Guerlet et al. (2014) tailored for Saturn with the icosahedral conditions CSP 2 and CSP 4 in the case of our Saturn DYNAMICO GCM simulations indicates that those are notfulfilled. / ◦ longitude / latitude, which made our reference simulation (sections 3 and 4); our Saturn DY-NAMICO GCM produces a satisfactory thermal structure, and seasonal variability thereof,compared to Cassini CIRS measurements (section 3.1).5. The number and intensity of mid-latitude eastward jets (and broad westward jets) reproducedby the reference GCM simulation is compliant with observations, although slightly under-estimated, but no stable circumpolar jet (less so hexagonal-shaped) is reproduced; thosejets’ intensities increase with altitude and their latitudinal organization exhibits potential-vorticity staircases (section 3.2.1).6. The GCM simulation exhibits at the equator both a superrotating zonal jet in the troposphereand stacked alternating zonal jets in the stratosphere; nevertheless, the former is one order ofmagnitude less powerful than the observed jet, and the latter are not downward-propagatingwith time as would be expected from the observed equatorial oscillation (section 3.2.2).7. Our model produces a wealth of Yanai (Rossby-gravity), Rossby and Kelvin waves in thetropical channel, part of them hinted at in available observations (section 3.3.1).8. Outside the tropics, the cores of eastward jets are perturbed by Rossby waves reminiscentof Ribbon-like waves, while westward jets host an eddy activity not especially organizedin vortices, which transitions in polar regions into a predominance of large-scale vortices(section 3.3.2).9. In the 15-Saturn-year course of our Saturn DYNAMICO GCM simulation, eastward jetsundergo poleward migration and perturbations by bursts of eddies (section 4.1).10. The global kinetic energy conversion rate simulated in our Saturn DYNAMICO GCM, albeithalf the value of the observed estimates, is positive and argues for a significant contributionof eddy acceleration in driving the eastward jets (section 4.2).11. The acceleration (and, if applicable, migration) of jets caused by eddy momentum trans-fers is evidenced by local analysis with either Eulerian-mean or transformed Eulerian-meanformalism (section 4.3).12. Eastward jets produced by our Saturn DYNAMICO GCM are prone to both barotropic andbaroclinic instabilities (section 4.4). 44ased on the present study, and the comparison between available observations and our GCMsimulations, we can envision the following improvements of our Saturn DYNAMICO GCM in thefuture:( a ) to refine the horizontal resolution to 1 / ◦ ;( b ) to include a physically-based parameterization for subgrid-scale dry and moist convection,and to subsequently deepen the model bottom to 10-20 bars;( c ) to extend the model top to the upper stratosphere and to refine the vertical resolution in thetroposphere and the stratosphere;( d ) to implement a parameterization for the impact of unresolved mesoscale gravity waves onthe mean flow.This list is not exhaustive, but represents the near-future evolution of our model. Long-termdevelopments would be inspired by the model-coupling methodology for the climate of telluricplanets, i.e. coupling our Saturn DYNAMICO GCM with stratospheric photo-chemical models,deep-interior convection models and upper-atmosphere thermo-iono-spheric models.What the refinement of horizontal resolution would bring to Saturn GCM studies can be il-lustrated by the tests of an earlier version of the Saturn DYNAMICO GCM at the horizontalresolutions of 1 / ◦ and 1 / ◦ . We carried out overdissipated GCM simulations at those resolutions,using both a sponge layer and hyperdiffusion timescales of the order τ D = 500 − ,
000 cores). Those simulations are thus not optimized for scientificreturn and the approach described in appendix A will have to be carried out again in the futurefor the 1 / ◦ and 1 / ◦ configuration of the model. Yet, even those imperfect 1 / ◦ and 1 / ◦ simula-tions with the Saturn DYNAMICO GCM demonstrate the potential of the model to better resolveeddies, waves and vortices with refined horizontal resolution, as is shown in Figure 20 and in themovie included as supplementary material, showing wind amplitude and vorticity from the 1 / ◦ Saturn DYNAMICO simulation.Finally, we note that the Saturn DYNAMICO GCM is only a first step towards a GCM systemable to simulate the atmospheres of all giant planets, ice giants Uranus and Neptune included.The development of the Jupiter DYNAMICO GCM is currently ungoing (Guerlet and Spiga, 2016;Boissinot et al., 2018), along the lines drawn for Saturn by Guerlet et al. (2014) and this study.The similarities and differences between Jupiter and Saturn in their banded jets (Ingersoll, 1990;Dowling, 1995; Kaspi et al., 2018), eddy activity (Salyk et al., 2006) equatorial oscillations (Li andRead, 2000; Simon-Miller et al., 2007), large-scale vortices (Youssef and Marcus, 2003; Fletcheret al., 2010b; Simon et al., 2014; Legarreta and S´anchez-Lavega, 2005) relate to fundamentalresearch in geophysical fluid dynamics. Employing GCMs for giant planets could help, along withobservations, to reach a detailed understanding of the big picture of giant planets’ climate andmeteorology. This is all the more relevant to prepare the next round of observations of the giant This is all the more true since the outcome of a 2-Saturn-year simulation at 1 / ◦ (hence not fully spun-up) wasused to initialize a 1 / ◦ simulation, and similarly from the 1 / ◦ to the 1 / ◦ simulations Instantaneous zonal winds predicted at 0.5 bar (yellow/red: prograde jets, blue/violet:retrograde jets) after simulated Saturn days by our Saturn DYNAMICO GCM. Our simulationsused a horizontal resolution of / ◦ (left) and / ◦ (right) longitude, and extent from troposphereto the stratosphere. In the left plot, jets’ instabilities and filamentation can be noticed; in the rightplot, the even finer horizontal resolution allows the model to reproduce the propagation of gravitywaves on the flanks of the jets, as well as the possible emergence of traveling vortices (cf. blue/greenspots in the northern hemisphere). The objectives behind those simulations were more technical(testing a massively-parallel computing cluster) than scientific: the strong hyperdiffusion made thestructures seen in the figure to disappear after about a thousand simulated Saturn days. planets, with either probes (Mousis et al., 2014), telescopes (Norwood et al., 2014), or orbitingspacecraft (Cavali´e et al., 2017). Acknowledgments
The authors thank the editor Darrell Strobel, an anonynous reviewer and Adam Showman forextremely constructive reviews which helped to improve the manuscript. We would like to thankTapio Schneider, Yohai Kaspi, Leigh Fletcher, Glenn Orton, Roland Young, Peter Read, MikeFlasar, Fran¸cois Forget, Michel Capderou, Pierre Drossart, Thibault Cavali´e, Agustin Sanchez-Lavega, Ricardo Hueso, Th´er´ese Encrenaz, Emmanuel Lellouch, Fr´ed´eric Hourdin, S´ebastien Fro-mang, for useful discussions and questions on preliminary versions of the work reported in thispaper.The authors acknowledge exceptional computing support from Grand ´Equipement National de46alcul Intensif (GENCI) and Centre Informatique National de l’Enseignement Sup´erieur (CINES).All the simulations presented in this paper were carried out on the
Occigen cluster hosted at CINES.This work was granted access to the High-Performance Computing (HPC) resources of CINES un-der the allocations 2015-017357, 2016-017548, A001-0107548, A003-0107548, A004-0110391 madeby GENCI. We also thank CINES for a “Grand Challenge” exceptional allocation in early 2015 totest the performances of the Saturn DYNAMICO GCM at various horizontal resolutions.Dubos, Cabanes, Spiga, Meurdesoif, Millour acknowledge funding from Agence Nationale de laRecherche (ANR), project HEAT ANR-14-CE23-0010. Guerlet, Indurain, Spiga acknowledge fund-ing from Agence Nationale de la Recherche (ANR), project OMAGE ANR-12-PDOC-0013. Spiga,Cabanes, Guerlet acknowledge funding from Agence Nationale de la Recherche (ANR), projectEMERGIANT ANR-17-CE31-0007. Guerlet, Spiga, Lebonnois acknowledge funding from Cen-tre National d’ ´Etudes Spatiales (CNES) project exploiting CIRS measurements onboard Cassini.Boissinot, Spiga acknowledge funding from Region ˆIle-de-France DIM ACAV+ project JOVIEN.Sylvestre, Fouchet, Spiga acknowledge funding from Universit´e Pierre et Marie Curie (now Sor-bonne Universit´e) ´Emergence program. Leconte acknowledges that this project has received fund-ing from the European Research Council (ERC) under the European Union’s Horizon 2020 researchand innovation program (grant agreement No. 679030/WHIPLASH).
A Impact of assumptions in the dynamical core
Numerical subgrid-scale dissipation in the model was found to be a critical setting to deal with.The sensitivity of simulated jets with horizontal dissipation is a common issue in GCM: it wasspecifically discussed for the cases of Venus (Lebonnois et al., 2013), Titan (Newman et al., 2011),and hot Jupiters (Thrastarson and Cho, 2011). This is a particularly important issue for a gas giantGCM, and it is discussed in section A.1. Another important question is to explore the behaviourof global axial angular momentum in our Saturn DYNAMICO GCM simulations, which is done insection A.2.As far as the sensitivity of the modeled jets to the settings adopted for bottom drag ( τ R , ϕ R )is concerned, we rely on the work by Liu and Schneider (2010) and Liu and Schneider (2011), andadopt their settings τ R = 100 Earth days and ϕ R = 33 ◦ for our reference simulation. Simulationswith Saturn DYNAMICO carried out with different values of those parameters (respectively τ R =10 Earth days and ϕ R = 10 ◦ ) confirm the conclusions of Liu and Schneider (2010) and Liu andSchneider (2011) that the bottom drag affects the jets’ width and speed. A.1 Exploring the impact of dissipation
A subgrid-scale dissipation term is included in our Saturn DYNAMICO GCM to prevent theaccumulation of energy at scales close to the grid resolution, caused by the GCM not resolvingthe turbulent scales at which this energy is dissipated. This hyperviscosity term is written in ourSaturn DYNAMICO model as an iterated Laplacian term on a given variable ψ (cid:34) d ψ d t (cid:35) dissip = ( − q +1 (cid:96) min τ ψD ∇ q ψ (16)47here q is the order of dissipation and τ ψD the damping timescale associated with the variable ψ atthe smallest spatial scale (cid:96) min resolved by the model for a given horizontal discretization. Largevalues of τ ψD means weaker dissipation: τ ψD is the times it takes to dissipate a perturbation onvariable ψ developing at the spatial scale (cid:96) min. The three variables denoted by ψ are vorticity,divergence, and potential temperature, chosen to set horizontal dissipation on respectively therotational component of the flow (e.g. Rossby waves), the divergent component of the flow (e.g.gravity waves), and the diabatic perturbations (e.g. coming from the physical packages). In GCMsfor telluric planets, strong variations from one grid point to one another result from topographycontrasts or mesoscale convective cells, which calls for a preferential damping on the divergent flow(Hourdin et al., 2012). In our Saturn DYNAMICO GCM simulations, we adopt a simpler approachand we set the same dissipation rate for all three variables, namely τ ψD ≡ τ D for divergence, vorticityand potential temperature.In the GCM methodology, suitable values of ( q, τ D ) are determined empirically, using a com-bination of past modeling experiences and trial-and-error approach using GCM simulations. The“right” settings for numerical dissipation is a trade-off between ensuring model stability, dampingenergy at the smallest resolved scales, and minimizing impact on the large-scale flow. A commonpractice is q ranging between 1 and 4, and τ D typically one-two terrestrial hours (2000 − / ◦ − ◦ GCM simulation. We chose q = 2 (fourth-order dissipation) for the Saturn DY-NAMICO GCM simulations, since it is the setting adopted by our team for GCM for telluricbodies (Hourdin et al., 1995; Lebonnois et al., 2010), and because q = 1 is overly dissipative oncirculations at large scales, while q = 3 led to results similar to q = 2. We then carried out severalone-Saturn-year simulations with our Saturn DYNAMICO GCM to explore the sensitivity of thecomputed tropospheric jet structure to the dissipation timescale τ D .Figure 21 shows that the simulated tropospheric jets in our 1 / ◦ Saturn DYNAMICO GCMsimulations are sensitive to the value assumed for dissipation timescale τ D . A first order-of-magnitude sensitivity study (Figure 21, top) using extreme values disqualifies the strongest dis-sipation rate ( τ D = 500) which damps the mid-latitude jets out of existence. Setting a weakdissipation ( τ D = 50000) lets jets significantly accelerate within one simulated year – before theGCM simulation undergoes numerical instabilities in the second year of simulation. A refined sen-sitivity study (Figure 21, bottom) indicates that results obtained with τ D = 10000 or τ D = 15000are essentially similar: the choice of dissipation mainly impacts jets’ meridional location.The selective criterion to choose τ D for our reference Saturn DYNAMICO GCM simulationis then based on observations of Saturn’s jets (Porco et al., 2005; Garc´ıa-Melendo et al., 2010;Studwell et al., 2018) from which we argue that, qualitatively, τ D = 10000 sets a more realisticvelocity profile. With τ D = 5000, high and mid latitudes jets are inconsistently weak, while τ D = 15000 smoothes the prevailing contrast that exists between equatorial and high-/mid-latitudejets. The value of τ D = 10000 is thus adopted for the 15-year-long reference simulation discussedin sections 3 and 4. It is important to note here that τ D = 15000 would have been an acceptablechoice as well. Figure 21 (right) shows that, even the intensity and location of jets can be differentbetween the Saturn DYNAMICO GCM simulations with either τ D = 10000 or τ D = 15000, theoverall jet structure is qualitatively similar in both cases.To understand why dissipation impacts the jets, it is important to keep in mind that eddies,which putatively drive the jets, have typical length scales of a couple degrees latitude for a grid48igure 21: Zonal-mean jets obtained at cloud level after integrated Saturn days (about onesimulated Saturn year) with our Saturn DYNAMICO GCM run at a horizontal resolution of / ◦ .Those tests were carried out with a preceding, slightly different, version of the DYNAMICO modelcompared to the one used for the reference simulations of section 3. In the top plot, results areshown for three values of horizontal dissipation timescales varying by one order of magnitude(orange: strong τ D = 500 s, blue: moderate τ D = 5000 s, green: weak τ D = 50000 s). In thebottom plot, results are shown for three values of horizontal dissipation timescales, two of them(blue: τ D = 5000 s, green: τ D = 15000 s) enclosing the value chosen for the reference simulationdiscussed in sections 3 and 4 (orange: τ D = 10000 s). . ◦ (assuming the eddies are baroclinically driven and taking the Rossby radius ofdeformation as a first-order theoretical estimate of the eddy length scale, e.g. Young and Read,2017). These eddies are thus potentially dampened by numerical dissipation that dominantlyacts on the smallest-resolved scales. Undoubtedly, the next challenge is to refine GCM resolutiontowards 1 / ◦ (or better) to enhance simulations of eddies, and hence planetary jets. Even atsuch refined horizontal resolution though, the chosen value of τ D will still be impacting the jets’intensities and locations (see also equation 21 in section A.2). Ultimately, how dissipative theSaturn atmosphere is for small-scale circulations remains an open question and difficult to constrainwith observations (Ingersoll et al., 2018). A.2 Angular momentum
The atmospheric Axial Angular momentum (AAM) M is defined by the sum of the planetarycontribution M m associated with the solid-body rotation of the planetary sphere and the relativecontribution M w associated with the motions of the atmosphere with respect to the solid-bodyrotating reference M = M m + M w = (cid:90) V µ m d m + (cid:90) V µ w d m (17)where (cid:82) V denotes global integration over the volume of the atmosphere and µ denotes the AAMterms per unit mass for respectively the planetary and relative contributions µ m = Ω a cos ϕ µ w = u a cos ϕ (18)Thus, the temporal evolution of AAM is either related to atmospheric mass redistribution (thatis, the temporal evolution of surface pressure) or wind variability (that is, the temporal evolutionof zonal wind). Assuming that hydrostatic primitive equations are considered, and in the absenceof any surface torque and zonal mechanical forcing, the globally-integrated AAM M is conservedprovided the top lid does not vary with longitude (Staniforth and Wood, 2003; Thuburn, 2008;Lauritzen et al., 2014), which is ensured by the DYNAMICO formulation (Dubos et al., 2015).The conservation of global AAM M in our Saturn DYNAMICO model is satisfying (Figure 22).There is a very small decrease of global AAM ( ∼ . ∼ . −
2% (reaching several tens of % for one particular model).In addition to the conservation of global AAM, negligible “AAM noise” must be ensured (Lebon-nois et al., 2012; Lauritzen et al., 2014). How the temporal evolution of global atmospheric AAMsplits between each term of the GCM tendencies can be writtend M d t = d M m d t + d M w d t = F + T + S + D + ε, (19)where F is the AAM tendency associated with subgrid-scale mixing in the physical packages(mostly boundary-layer effects), T is the AAM tendency associated with mountain torques, S is50igure 22: Temporal evolution of the global AAM M (normalized to the initial value M =5 . × kg m s − ) for the complete duration of our reference Saturn GCM simulation. the AAM tendency associated with upper sponge layer (a reminder that upper-level sponge layermight alter significantly the AAM balance, Shaw and Shepherd, 2007), D is the AAM tendencydue to conservation errors in the parameterization of horizontal dissipation, and ε is a residualnumerical rate of AAM variation due to conservation errors (hereinafter named the “AAM noise”since it is a spurious source/sink of AAM in the model). The AAM noise ε can be diagnosed ina GCM by adding the temporal variations of M m and M w computed by the primitive equations( P E ) in the dynamical core, excluding any term accounted for in F , T , S or D (e.g. explicitdiffusion operators are included in D ) ε = (cid:34) d M m d t (cid:35) P E + (cid:34) d M w d t (cid:35) P E (20)In the impossible perfect situation where no numerical approximations or errors are made in theGCM dynamical core when solving the primitive equations, the AAM noise ε should be zero sinceAAM is exactly transferred from the solid-body rotation to the atmospheric flow and vice versa;a GCM simulation where ε is of similar or larger magnitude than the other torques would bequestionable (Lauritzen et al., 2014).In our Saturn GCM simulations with DYNAMICO, D also includes the Rayleigh friction at thebottom of the atmosphere, S is zero because no upper-level sponge layer is used, T is zero sincegas giants are devoid of surface, and for a similar reason, in practice the term F is one order ofmagnitude smaller than other terms. This means that in the case of our Saturn GCM (and moregenerally for any gas giant GCM) equation 19 reduces tod M d t = d M m d t + d M w d t = D + ε (21)Hence the angular momentum noise ε has potentially a strong impact on the temporal AAM51ariability in a gas giant GCM. A similar concern is raised by Lebonnois et al. (2012) in thecase of idealized Venus simulations without topography. The requirement of low AAM noise isless stringent in realistic GCM simulations for Earth, Venus, or Titan where the contribution ofmountain torques T is large (Lebonnois et al., 2012; Lauritzen et al., 2014).Figure 23: Temporal evolution of the tendency d M / d t of global AAM for the complete durationof our reference Saturn GCM simulation. Blue points depict the term [ d M w / d t ] P E associated withwind tendencies computed in the primitive equations resolved in the dynamical core of the SaturnGCM. Red points depict the term ε associated with AAM noise, i.e. residual numerical rate ofAAM variation due to conservation errors. Green points depict the term D associated with AAMtendencies resulting in the dissipation and Rayleigh drag scheme. The log-scale figure is producedwith the absolute values of the various tendencies d M / d t . Compared to the analysis presented in Lebonnois et al. (2012) and Lauritzen et al. (2014) fortelluric planets, the analysis of AAM noise is more straightforward for gas giants. The following52ondition must be ensured ε (cid:28) (cid:34) d M w d t (cid:35) P E (22)Figure 23 shows that this condition 22 is fulfilled in our Saturn GCM, with AAM noise being one totwo orders of magnitude smaller than AAM tendencies computed in the dynamical core’s primitiveequations (except for a short duration close to one and half simulated year). This indicates thatAAM noise does not alter the dynamical predictions for jets and eddies. The same conclusionstands for the AAM tendencies associated with dissipation and Rayleigh drag (Figure 23), whichare even smaller than the AAM noise. This indicates that the horizontal dissipation set in ourGCM (section A.1) has a negligible impact on the AAM budget – this is consistent with the overalljet structure being similar in simulations using either τ D = 10000 or τ D = 15000.In an attempt to explore the model settings that could influence the AAM noise ε , we found thatit is basically insensitive to time step, vertical discretization, horizontal dissipation (only a verystrong horizontal dissipation significantly lowers AAM noise, but adversely impacts the intensityof jets). This is consistent with Lauritzen et al. 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