Global climate modeling of Saturn's atmosphere. Part IV: stratospheric equatorial oscillation

The Composite InfraRed Spectrometer (CIRS) on board Cassini revealed an equatorial oscillation of stratospheric temperature, reminiscent of the Earth's Quasi-Biennial Oscillation (QBO), as well as anomalously high temperatures under Saturn's rings. To better understand these predominant features of Saturn's atmospheric circulation in the stratosphere, we have extended towards higher altitudes the DYNAMICO-Saturn global climate model (GCM), already used in a previous publication to study the tropospheric dynamics, jets formation and planetary-scale waves activity. Firstly, we study the higher model top impact on the tropospheric zonal jets and kinetic energy distribution. Raising the model top prevents energy and enstrophy accumulation at tropopause levels. The reference GCM simulation with 1/2$^{\circ}$ latitude/longitude resolution and a raised model top exhibits a QBO-like oscillation produced by resolved planetary-scale waves. However, the period is more irregular and the downward propagation faster than observations. Furthermore, compared to the CIRS observation retrievals, the modeled QBO-like oscillation underestimates by half both the amplitude of temperature anomalies at the equator and the vertical characteristic length of this equatorial oscillation. This QBO-like oscillation is mainly driven by westward-propagating waves; a significant lack of eastward wave-forcing explains a fluctuating eastward phase of the QBO-like oscillation. At 20$^{\circ}$N and 20$^{\circ}$S latitudes, the DYNAMICO-Saturn GCM exhibits several strong seasonal eastward jets, alternatively in the northern and southern hemisphere. These jets are correlated with the rings' shadowing. Using a GCM simulation without rings' shadowing, we show its impact on Saturn's stratospheric dynamics. Both residual-mean circulation and eddy forcing are impacted by rings' shadowing.


Introduction
The longevity of the Cassini mission permitted an unprecedented spatial and seasonal coverage of Saturn's stratosphere. In particular, the Composite InfraRed Spectrometer (CIRS) instrument on board Cassini revealed stratospheric phenomena analogous to ones occuring in Earth's and Jupiter's stratospheres (Dowling, 2008). Firstly, seasonal monitoring of hydrocarbons in Saturn's stratosphere suggests a conceivable inter-hemispheric transport of stratospheric hydrocarbons (Guerlet et al., 2009Sinclair et al., 2013;Fletcher et al., 2015;Sylvestre et al., 2015), similar to the Earth's Brewer-Dobson circulation which affects the stratospheric ozone distribution (Murgatroyd and Singleton, 1961;Dunkertton, 1979;Solomon et al., 1986;Butchart, 2014). The Cassini mission further revealed a lack of temperature minimum under the rings' shadow (that was expected by the radiative balance Fletcher et al. (2010); Friedson and Moses (2012); Guerlet et al. (2009Guerlet et al. ( , 2010Guerlet et al. ( , 2014), which is an additional hint of subsidence motion in the winter hemisphere.
Secondly, a crucial discovery of Cassini is that temperatures retrieved from thermal infrared spectra in Saturn's stratosphere exhibit an equatorial oscillation with semi-annual periodicity (Orton et al., 2008;Fouchet et al., 2008).
During the 13-Earth-years cruise of Cassini around Saturn, CIRS and radio oc-cultations measurements permitted to characterize this equatorial oscillation of temperature and its downward propagation over seasonal timescales (Guerlet et al., 2011;Li et al., 2011;Schinder et al., 2011;Guerlet et al., 2018). Alternatively eastward and westward stacked jets are associated with the temperature signatures detected by Cassini, according to the thermal wind equation.
This oscillation is called Saturn "Quasi-Periodic Oscillation" or "Semi-Annual Oscillation", due to its half-Saturn-year periodicity. Equatorial oscillations appear to be common phenomena in planetary stratospheres. Observations of Jupiter's stratosphere revealed an equatorial oscillation of temperature (Leovy et al., 1991;Orton et al., 1991) associated with a wind propagation reversal (Friedson, 1999;Simon-Miller et al., 2007). This oscillation exhibits a period of 4.4 Earth years and has been designated as the jovian Quasi-Quadrennial Oscillation. Equatorial oscillations are also suspected on Mars (Kuroda et al., 2008;Ruan et al., 2019). Those equatorial oscillations in planetary atmospheres are reminiscent of the Earth's stratospheric Quasi-Biennial Oscillation (QBO) and Semi-Annual Oscillation (SAO) (Lindzen and Holton, 1968;Andrews et al., 1983;Baldwin et al., 2001).
Earth's Quasi-Biennial Oscillation (QBO) results from wave-mean flow interactions (Reed et al., 1961;Andrews et al., 1983;Baldwin et al., 2001). The QBO is driven by the vertical propagation of tropospheric waves -both planetaryscale and mesoscale waves -to stratospheric altitudes where they break either by becoming convectively unstable or by encountering critical level (depending on vertical and horizontal wavelengths, wave phase speeds and vertical shear (Lindzen and Holton, 1968)). Equatorially trapped waves carry eastward and westward momentum, as well as warm and cold temperature anomalies. During wave breaking, momentum and heat are transferred to the mean flow, changing the large-scale zonal wind and temperature fields. The induced large-scale thermal perturbations, and zonal wind reversal, oscillate with an average period of 28 months and a mean downward propagation rate of 1 km per month on Earth (Reed et al., 1961;Mayr and Lee, 2016). The eastward and westward phases of the Earth's equatorial oscillation are of similar amplitude and are created by a diversity of eastward-and westward-propagating wave sources (Dunkerton, 1997;Hamilton et al., 2001;Giorgetta et al., 2002;Ern and Preusse, 2009).
The terrestrial QBO's eastward phase is produced by the wave breaking of eastward-propagating gravity waves (from tropospheric moist convection) and Kelvin waves. Kelvin waves contribute to about 30% of the eastward forcing (Ern and Preusse, 2009), whereas mesoscale gravity waves contribute to 70% of the eastward forcing (Dunkerton, 1997). The westward phase of the Earth's QBO is induced by Rossby and mixed Rossby-gravity waves' breaking, as they carry westward momentum. Furthermore, inertia-gravity waves transport eastward and westward zonal momentum, thus are involved in both the eastward and westward phases of the QBO (Baldwin et al., 2001).
Despite differences in their periods, Jupiter's and Saturn's equatorial oscillations share similarities with the Earth's (Dowling, 2008), which raises questions about the driving mechanisms of the gas giants' equatorial oscillations.
Building on tools developed throughout the long history of Earth's atmospheric modeling, numerical models of global circulation on Jupiter and Saturn have been developed for almost 20 years. Stratospheric oscillations on gas giants have been previously addressed by a handful of those models. Three studies employed a quasi-two-dimensional modeling framework that only resolves meridional and vertical structure and parameterize longitudinal forcing. On the one hand, Li and Read (2000) demonstrated that the major contribution to a QBO-like oscillation in Jupiter's atmosphere was planetary-scale waves (Rossby, mixed Rossby-gravity and Kelvin waves). On the other hand, Friedson (1999) and Cosentino et al. (2017) showed that a parameterization of mesoscale gravity waves enabled to reproduce the observed jovian equatorial oscillation -with no need to invoke planetary-wave forcing.
Regardless of the nature of waves invoked in these studies, QBO-like oscillations are phenomena resulting from wave propagation in longitude, latitude and on the vertical that cannot be fully resolved by 2D models: the 3D propagation of waves, and their impact on global circulation, must be parameterized in 2D models. Thus, to better understand the wave-mean flow interactions leading to the observed gas giants' equatorial oscillations, atmospheric models must solve the three-dimensional dynamics. A fully three-dimensional model to study in detail the global troposphere-to-stratosphere circulation in gas giants has only been recently developed, because the huge computational resources required to resolve eddies arising from hydrodynamical instabilities that putatively force equatorial oscillations. Showman et al. (2018) were the first to show the development of a QBO-like oscillation in an idealized 3D global primitive-equation model for gas giants and brown dwarfs. Their model uses a random wave forcing parametrization at the radiative-convective boundary to drive equatorial oscillations, and a Newtonian scheme to represent radiative heating/cooling in the thermodynamics energy equation. A stack of eastward and westward jets that migrate downward over time is created in the stratosphere of Showman et al.
(2018)'s model, analogous to Earth's QBO, with a range of periods similar to Jupiter's and Saturn's equatorial oscillations. Nevertheless, the QBO-like oscillations depicted in their model occur at lower pressure levels (between 10 5 and 10 3 Pa) than the observations (Saturn's equatorial oscillation extends from 10 3 to 1 Pa).
To gain further insights on Saturn's equatorial oscillation, this work aims at adopting the three-dimensional approach of Showman et al. (2018) with a more realistic representation of radiative transfer and wave dynamics in Saturn's stratosphere. The present paper is part IV of a series of papers about global climate modeling of Saturn. In part I, Guerlet et al. (2014) built a complete seasonal radiative model for Saturn. In part II, Spiga et al. (2020) coupled this radiative model to a new hydrodynamical core (Dubos et al., 2015) to obtain the DYNAMICO-Saturn Global Climate Model (GCM). Using this GCM tailored for Saturn, Spiga et al. (2020) simulated Saturn's atmosphere from the troposphere to the lower stratosphere for 15 Saturn years at fine horizontal resolution, without any prescribed wave parametrization. The DYNAMICO-Saturn GCM simulation described in Spiga et al. (2020) produced consistent thermal structure and seasonal variability compared to Cassini CIRS measurements, mid-latitude eddy-driven tropospheric eastward and westward jets commensu-rate to those observed (and following the zonostrophic regime as is argued in the part III paper by Cabanes et al. (2020)), and planetary-scale waves such as Rossby-gravity (Yanai), Rossby and Kelvin waves in the tropical channel. While the simulations in Spiga et al. (2020) exhibited stacked eastward and westward jets in the equatorial stratosphere, those jets were not propagating downwards, contrary to the observed equatorial oscillation. The likely reason for this is that the model top was too low and the vertical resolution too coarse to address the question of a QBO-like oscillation in the stratosphere.
In this paper, we use the DYNAMICO-Saturn model as in Spiga et al. (2020) with a top of the model extended to the higher stratosphere. We describe here the benefits on the stratospheric dynamics of using a wider vertical extent in our GCM. In section 2, we describe briefly our DYNAMICO-Saturn GCM. Section 3 presents, through comparison between Spiga et al. (2020) and our results, the impact on tropospheric dynamics of raising the model top. Section 4 focuses on the stratospheric dynamics and wave-mean flow interactions producing a QBO-like oscillation in our Saturn GCM. In addition, a strong seasonal wind reversal is obtained at 20N and 20S. We study this phenomenon and discuss the impact of the rings' shadowing on the stratospheric tropical dynamics in section 5. Summary, conclusions and perspectives for future improvements are explained in section 6.

DYNAMICO-Saturn
A complete description of the DYNAMICO-Saturn is available in Spiga et al. (2020). A Global Climate Model (GCM) is composed of two parts: a dynamical core to resolve the Navier-Stokes equations on planetary scales and a physical package which is an ensemble of parametrizations to describe sub-grid-scale processes. The Saturn GCM developed at Laboratoire de Météorologie Dynamique employs DYNAMICO, a dynamical core using an icosahedral grid that ensures conservation and scalability properties in massively parallel resources (Dubos et al., 2015). It solves the primitive hydrostatic equations assuming a shallow atmosphere. The dynamical core is bounded by a simple bottom Rayleigh drag (Liu and Schneider, 2010) to account to zeroth order for deep interior phenomena (see section 2.2 of Spiga et al., 2020). To conserve angular momentum, DYNAMICO-Saturn does not use an absorbing layer at the bottom of the model (see Spiga et al. (2020)'s Appendix).
The physical package used in our DYNAMICO-Saturn is tailored for Saturn, particularly regarding radiative transfer (this is fully detailed in the part I paper by Guerlet et al. (2014)). Radiative transfer computations use correlated-k ta- Most simulation settings in this part IV paper are analogous to those adopted in the part II paper by Spiga et al. (2020). For our reference simulation, we employ Saturn-DYNAMICO with an approximate horizontal resolution of 1/2 • in longitude/latitude. The time step of calculations is 118.9125 seconds, with physical packages called every half a Saturn day and radiative computations done every 20 Saturn days, because of the long radiative timescales in troposphere and stratosphere of gas giants. The major difference between Spiga et al.
(2020) (Part II) and the present Part IV reference simulation is the inclusion of 61 levels in the vertical dimension, extending from the troposphere p bottom = 3.10 5 Pa (or 3 bars) to the upper stratosphere p top = 10 −1 Pa (1 µbar).
Simulations using our DYNAMICO-Saturn are initialized at every horizontal grid point with a vertical profile of temperature computed by a 1D radiativeconvective equilibrium model of Saturn's atmosphere (Guerlet et al., 2014), using the same physical parametrizations than our DYNAMICO-Saturn GCM.
The single-column simulation starts with an isothermal profile and runs for twenty Saturn years to reach the annual-mean steady-state radiative-convective equilibrium. The initial zonal and meridional winds are set to zero in our simulations. DYNAMICO-Saturn simulations requires radiative and dynamical spin-up of about five years for the tropopause levels to ensure a dynamical steady-state . Hence, in what follows, we present 12-Saturn- year-long simulations to study Saturn's stratospheric dynamics.
3. Impact of the extended model top on Saturn's tropospheric dynamics In this section, we present a comparison of tropospheric jets as simulated by Spiga et al. (2020) versus the present work, to check that the tropospheric dynamical regime is conserved with the new vertical grid. Figure 1 displays altitude/latitude sections of zonal-mean zonal wind after 11 years of simulation for each vertical grid. Both simulations present similar results in the troposphere, although with some notable differences. The 32-vertical-level simulation ) exhibits about 10 jets extending from 3.10 5 Pa (or 3 bars) to the model top (10 2 Pa or 1 mbar). At the equator, there is a stacking of eastward and westward jets between 2.10 4 Pa and the model top. With an extended vertical range (model top at 10 −1 Pa), we now distinguish 14 jets into the tro-posphere and 9 jets in the stratosphere. The equatorial stacking of wind seen in 32-level simulation is now clearly localized between 2×10 2 Pa (tropopause) and the model top. Contrary to previous GCM works Showman, 2008, 2010;Schneider and Liu, 2009), we remark that the eastward wind patterns of the 61-level simulation decay with altitude, which is not the case in the 32-level one (except for the equatorial jet). Thus the behavior of our 61-level troposphere-to-stratosphere simulation is consistent with observations (Fletcher et al., 2019). We also note that the equatorial tropospheric eastward jet is less intense in the reference (61-level) simulation than in the 32-level one.
We performed a comparative analysis of the kinetic energy distribution in   Cabanes et al. (2020)) and the reference simulation (right). All spectra are averaged in time between the tenth and the twelfth simulated Saturn years, and over two adjacent vertical levels in the troposphere. The zonal energy spectrum, that corresponds to the axisymmetric part of the flow (i.e. the jets, displayed with red lines), follows the theoretical -5 slope coined by Sukoriansky et al. (2002), on the range of latitudinal wavenumbers n ∼ 10 − 100. At large scale (i.e. small wavenumbers n), zonal energy spectra are two orders of magnitude higher than the residual energy spectra (black solid lines) that stand for the non-axisymmetric flow component. Flow anisotropy between the axisymmetric and non-axisymmetric components can be related to the regime of zonostrophic turbulence (Galperin et al., 2010). In both simulations, the energetic maximum is well estimated by the Rhines typical length scale n R = R(β/2U ) 1/2 that set typical jets' size (Rhines, 1977) In the 32-level simulation of Spiga et al. (2020), the residual spectrum barely fits a -5/3 slope evocative of an inverse energy cascade as predicted by the theory of rotating turbulence (Kraichnan, 1967a;Sukoriansky and Galperin, 2016).However, the −5/3 slope does not predict the residual spectrum in the 61-level reference simulation, showing that the classical inverse cascade scenario doesn't apply to our 3D simulation. By raising the model top, we significantly impact the large scale dynamics beyond the Rhines scale(n < n R ). In 32-level simulation, very energetic large scale features at wavenumbers n < n R show that the dynamics is dominated by m = 1, 2 and 3 non-axisymmetric modes.
On the contrary kinetic energy strongly decrease at those scale for the 61-level simulation. Then, it is likely that the energetic m = 1, 2 and 3 modes enforced in the 32-level simulation are artefacts of the geometrical flow confinement in depth. Nonetheless, this can only be disclosed by computing similar statistical analysis from Saturn's direct observations to confirm our predictions.
Middle and bottom panels in Figure 2 show spectral energetic and enstrophy fluxes. Positive fluxes corresponds to downscale energy transfers (a "direct" cascade) and negative fluxes are upscale energy transfers (a "inverse" cascade).
In the range n < 10 2 , energy fluxes are clearly negative meaning that the atmospheric large scale flow is driven by an upscale energy cascade. As shown by Cabanes et al. (2020), this inverse cascade results from the barotropization of baroclinic eddies at small scales. The energetic power of the inverse cascade reaches 6×10 −6 W kg −1 for the 32-level simulation and 4×10 −6 W kg −1 for the 61-level simulation. With a similar energetic forcing, but with a larger extension in depth of model, the 61-level simulation reports a lower energetic power than the 32-level simulation at the same vertical level. Both estimates are however coherent with previous estimate for Jupiter, 9×10 −5 W kg −1 computed from direct observations from Cassini images (Young and Read, 2017).
Enstrophy fluxes are nearly similar in both simulations, they show a strong direct cascade at small scale and a tendency for an upscale cascade near the Rhines scale. In the present study, time-averaged spectral quantities appear to be in good agreement with the energy-enstrophy double cascade scenario that may be reminiscent of the double cascade described in 2D flows by Kraichnan (1967a) theory. Hence, although we find some differences, the overall dynamical regime in the troposphere is conserved when we raise the model top of our DYNAMICO-Saturn GCM towards stratospheric levels.
We now turn to the impact of the model top on the zonostrophic dynamical regime in the lower stratosphere. Figure  To summarize the conclusions of this section, global dynamics in Saturn's   troposphere is broadly similar in the 32-level simulation of  and the 61-level simulation presented here, with the latter exhibiting a more realistic quantitative energy and enstrophy fluxes transfers from the troposphere to the stratosphere.Raising the model top has made it possible to eliminate the energy accumulation in the lower stratosphere, while the zonostrophic dynamical regime and the jets' formation are analogous to the previous Spiga et al. (2020) Saturn GCM simulation. We now turn to the main goal of this paper, the stratospheric dynamics of Saturn's atmosphere.

Global stratospheric zonal wind
The temporal evolution of the zonal-mean stratospheric zonal wind over the whole 12-year duration of our DYNAMICO-Saturn simulation is shown in Figure 5 at the pressure level of 40 Pa. Before turning to low latitudes, we note that at high and mid-latitudes, zonal jets undergo a poleward migration due to bursts of eddies induced by baroclinic instabilities. This migration was already witnessed by Spiga et al. (2020) in the troposphere and is thought to be partly caused by baroclinicity at the bottom of the model (3×10 5 Pa), where the meridional gradient of temperature is slightly overestimated compared to the observations. Now, turning to the low latitudes simulated in our reference 61-level Saturn simulation, we notice in Figure 5 two key features in the temporal evolution of the stratospheric zonal-mean zonal wind: • an alternatively eastward and westward wind direction at the equator, with a sub-annual periodicity, appearing in the second simulated year and maintained for the rest of the 12 simulated years; • an alternatively eastward and westward wind direction at 20 • N and 20 • S, with an annual periodicity and a phase opposition between the northern and the southern hemisphere.
These two features are discussed in detail in the following sections.

Equatorial stratospheric zonal jets
By making an altitude/time section of the zonal-mean zonal wind at the equator ( Figure 6), for the four last years of our 12-years simulation, the downward propagation with time of the stacked stratospheric eastward-westward jets is well visible. At a given pressure level, the zonal wind alternates between eastward and westward direction, with a westward phase more intense -about -100 m s −1 (Figure 9 in Showman et al. (2018) shows a similar behaviour) -than the eastward one (only 60 m s −1 ). In addition, at a given pressure level, the westward phase lasts longer than the eastward one. The equatorial eastward phases of the oscillation seem to be unstable and disturbed compared to westward phases. For this reason, the period of the resulting equatorial oscillation is irregular in time.
As is reminded in the introduction, a pattern of alternatively eastward and  tions.
The second notable feature in Figure 5 is the strong alternating eastward and westward jets at around 20 • N and 20 • S, with a seasonal phase opposition between the northern and southern hemispheres. Compared to a time evolution of the incoming solar radiation (Figure 7), we remark that each eastward jets emerge during the winter and seem to be correlated to the rings' shadowing.
Those tropical eastward and westward jets both have an amplitude of about 100 m s −1 , contrary to the QBO-like equatorial oscillation which exhibits an eastward phase weaker than the westward phase. This pattern exhibits a regular one-Saturn-year periodicity without downward propagation: the jets shown in Figure 5 at 40 Pa extend from 10 3 Pa to the model top (Figure 8).

Equatorial stratospheric thermal structure
In this section, we compare our DYNAMICO-Saturn simulations with the observations of Saturn's equatorial stratosphere by Cassini/CIRS. Stratospheric winds have never been measured; rather, the CIRS instrument on-board the To compare our simulations to CIRS observations, we re-mapped altitude/latitude   We can also compare our simulations to the idealized work of Showman et al. (2018). In their idealized simulation, they obtain a temperature oscillation of   Another feature of Saturn's stratosphere observed by Cassini/CIRS is the anomalously high temperatures under the rings' shadows .
Previous radiative simulations of Saturn's atmosphere failed to reproduce these temperature anomalies (Guerlet et al., 2014), hence dynamical heating was considered to explain this feature. In our simulations, there is an interplay between the tropical eastward jets and temperature evolution, displayed in Figure 13.
In wintertime, at 20 • latitude, the temperature starts by cooling rapidly (from 150K to 140K), which is likely due to the ring's shadow radiative effect. Then, when the eastward jet increases in strength, this low temperature region is significantly reduced, from 140K to 145K. This is true only near the core of the eastward jet, at 20 • . Indeed, at 30 • latitude, the temperature remains very cold (140K) throughout wintertime. We will assess in more detail the dynamical impact of ring shadowing in the stratosphere in section 5.

Stratospheric kinetic energy distribution of the wind field
Using the tools described in section 3, we realize an analysis of the kinetic energy distribution of the stratospheric zonal wind to determine the dynamical regime related to the QBO-like oscillation. The zonal kinetic spectrum (red), the residual kinetic spectrum (black), the modal kinetic spectrum for m = 1 to

Eddy-to-mean interactions driving the equatorial stratospheric oscillation
To determine the eddy-to-mean interactions within the flow, we use the Transformed Eulerian Mean (TEM) formalism. In terrestrial atmospheric studies, it is commonly employed to investigate momentum and heat transfers by wave-mean flow interactions (Andrews et al., 1983).
In all the following equations (1 to 6), the overline denotes the zonal-mean of each field, the prime symbol denotes the eddy component, i.e. departures from the zonal mean. In equations 3 and 4, the red terms are the residual-mean circulation contributions and the blue terms are the eddy forcing in momentum and heat. By defining the transformed zonal-mean velocities as follows we obtain the momentum and energy equations in the Transformed Eulerian Mean formalism: Because Saturn is a fast-rotating planet, it is close to quasi-geostrophic equilibrium. In this case, the inertial acceleration, the Coriolis force and the pressure gradient are dominant and a zeroth-order balance exists between the three terms. Using this quasi-geostrophic approximation, we can simplify the Eliassen-Palm formulation to neglect the mid-latitude eddy heating term (the v θ term in the F φ component) and the meridional component of the zonal wind divergence (the 1 a cos φ ∂(u cos φ) ∂φ term in the F P component) which is important at high latitudes and negligible at low latitudes. We keep the vertical eddy momentum transport due to the equatorial waves (the u ω term in F P ), which is an important contribution in the wave forcing in the tropics (our analysis is focused around ±20 • in latitude). We thus define a simplified Eliassen-Palm flux as follows for the analysis of the subtropical regions: We thus obtain that the horizontal component of the simplified Eliassen-Palm (F φ s ) flux is anti-parallel to the meridional eddy momentum transport and the vertical component of the simplified Eliassen-Palm flux (F P s ) is proportional to the meridional eddy heating flux and anti-parallel to the vertical eddy momentum transport due to the equatorial waves. In the following analysis, we separate the lower stratosphere (2×10 4 Pa to 6×10 1 Pa) and the upper stratosphere (6×10 1 Pa to 5×10 −1 Pa) to clarify the analysis. For each stratospheric regions, the two components of the Eliassen-Palm vectors are scaled by their own average value which are not the same for the two stratospheric regions.
We ran an additional, specific 1000-day-long GCM simulations with a daily output frequency, restarted at a time of a strong reversal of wind direction from the westward to the eastward direction (around 8.4 simulated Saturn years).   the previously-mentioned westward acceleration due to eddies -extending from 20 Pa to the top of this figure. To summarize, this figure shows that waves induce eastward and westward accelerations just below the eastward and westward equatorial jets, respectively, that produce the downward propagation of the stacked jets causing the QBO-like oscillation.
5. Impact of the rings on Saturn's stratospheric dynamics

Zonal wind in the tropical regions
To determine Saturn's rings contribution to the atmospheric dynamics in the tropical channel, we performed an alternate simulation named the "no-ring" simulation, in which the rings' shadowing of the incoming solar radiation is neglected, all other settings being equal with the DYNAMICO-Saturn reference simulation described previously. This simulation starts with an initial state derived from this reference simulation after 8 simulated Saturn years. In this alternate simulation, the DYNAMICO-Saturn GCM was run for four simulated years: two years to reach a dynamical steady-state and two others to compare the results with the eleventh and the twelfth year of our reference simulation. Figure 18 compares the two Saturn GCM simulations. In the no-ring simulation compared to the reference simulation, we note a disturbance of the structure of the zonal-mean zonal wind at the tropics, as well as an enhancement of the zonal wind intensity at the equator. Eastward jets at the tropics in the no-ring simulation split in several weaker eastward jets in the northern hemisphere. In the southern tropics, eastward jets even disappear after 3.5 Saturn years of simulation. Hence, rings' shadowing (or the absence thereof) impacts equatorial and tropical dynamics.    limb observations (Guerlet et al., 2009). This comparison is carried out at Ls∼300 • in southern summer, when rings shadowing occurs at 20-25 • N. As mentioned previously, the radiative-convective model fails to reproduce the local temperature maximum under the rings' shadow (between 20 and 30 • N).

Temperature field in the tropical region
It predicts instead a temperature ∼15 K colder than the CIRS observations.
Besides, the reference GCM simulation presents high temperatures at these latitudes which are ∼10 K too warm compared to measurements. At northern mid-latitudes in the winter hemisphere, temperature decreases from 30 • N to 40 • N which is in agreement with the CIRS observations and was not predicted by the radiative-convective model. The no-ring simulation also exhibits warm temperature between between 20 • N and 30 • N; however, these temperatures are colder than the reference simulation (probably because the tropical eastward jets are impacted by the lack of ring shadowing) and still too warm compared to the observations. Hence Figure 19 suggests a possible dynamical origin of the unexpected high temperature below Saturn's rings.
At the equator, Figure 19 shows that the no-ring simulation and the radiativeconvective simulation temperatures are consistent with the CIRS observations.
On the contrary, the reference simulation underestimates the temperature at the equator by around 5 K. The presence of the rings impacts the equatorial temperature and the QBO-like oscillation. The tropic-to-equator meridional gradient of temperature is thus decreased in the reference simulation (compared to observations). According to the thermal wind balance, the vertical shear of the zonal wind is decreased too. As a result, the intensity and periodicity of the equatorial oscillation are impacted. The seasonality of the wave activity causes seasonality of eddy-to-mean interaction. We map the time evolution of the acceleration due to eddies ( Figure   22) in the reference simulation (top row) and the no-ring simulation (bottom row). In the reference simulation, eddy-induced acceleration is enhanced at the tropics when eastward tropical jets are present. There is a strong deceleration due to eddies at the location of the tropical eastward jets (except for the jets located at 20 • S at 2.5 simulated Saturn year). The winter tropics, where the rings' shadow occurs and eastward jets are produced, are associated with an intense interaction between eddies and the mean circulation, which induces a significant deceleration due to eddy activity. Regarding the no-ring simulation, the eddy-induced acceleration is halved compared to the reference simulation at the location of the eastward jets. The seasonality of the eddy-to-mean interaction persists at the tropics without the rings shadowing, but the intensity of the eddy forcing in the mean flow is reduced. In this case, the whole tropical channel is driven by homogeneous absolute values of acceleration and deceleration (about 1.5×10 −5 m s −2 ).

Eddy-to-mean interactions in the tropical region
To summarize the conclusions of this section 5, rings' shadowing affects the stratospheric dynamics of Saturn. The tropical thermal structure changes when ring shadowing is not considered. This has an impact on the mean meridional circulation and the eddies, which both force the tropical eastward jets. As a result, seasonally-varying tropical eastward jets are caused by ring shadowing.
A putative impact of the ring's shadow on Saturn's stratospheric dynamics  With our study, we show that such interpretations of anomalies seen in observations invoking upward and downward motions may reflect a too simplistic view. Rather, eddy forcings might play a role as much as significant as the residual-mean circulation in shaping Saturn's stratospheric temperatures and dynamics.

Conclusions and perspectives
We use a troposphere-to-stratosphere Saturn GCM without any prescribed wave parameterization to study the equatorial stratospheric dynamics, especially the stratospheric equatorial oscillation.
Our DYNAMICO-Saturn GCM reproduces an almost semi-annual equatorial oscillation with contrasted eastward and westward phases. This oscillation shows a similar behavior in temperature than the Cassini/CIRS retrievals, with alternatively local maxima and minima of temperature stacked on the vertical at the equator. The DYNAMICO-Saturn signal exhibits a two times smaller vertical characteristic size of the oscillation and underestimates by a factor of two the amplitude of the temperature anomalies, compared with CIRS observations.
Regarding the zonal-mean zonal wind, we determined an irregular period of wind reversal and a downward propagation rate faster than observations. Spectral analysis at the equator demonstrated that this QBO-like oscillation is produced by planetary-scale waves. The equatorial oscillation in the DYNAMICO-Saturn GCM is mainly driven by strong westward-propagating waves, such as Rossby, Rossby-gravity and inertia-gravity waves, which deposit westward momentum in the stratosphere. There are only two eastward-propagating modes (Kelvin waves) in the spectral analysis with a weaker impact in the stratospheric dynamics than the westward-propagating waves. This lack of eastward waves prevents the eastward momentum deposition in the mean zonal wind and explains the erratic behavior of the eastward phase of the modeled QBO-like oscillation.
Using the Transformed Eulerian Mean formalism to determine how the eddyto-mean interactions drives the Saturn equatorial oscillation, we are able to conclude that the maximum of eddy forcing comes from the high-troposphere tropical regions.
Moreover, we demonstrate the wave forcing origin of the vertically-stacked stratospheric eastward and westward jets that propagate downward with time to form the equatorial oscillation. At the equator, in the stratosphere, the eddyinduced eastward acceleration maximum is located just under the eastward jets and the eddy-induced westward acceleration maximum is located just under the westward jets at every step of the downward propagation of this Saturn QBO-like equatorial oscillation.
For future improvements of the modeling of the QBO-like oscillation in Saturn's equatorial stratosphere, we will draw inspiration from Earth's atmospheric modeling. Adequate vertical resolution is needed to obtain a more realistic Quasi-Biennial Oscillation in Earth models (Richter et al. (2014). Refining the vertical resolution in the stratosphere both leads to a downward propagation rate of the oscillating zonal wind consistent with the observations, and a large-enough amplitude of Kelvin and Rossby-gravity waves to enhance the westward and eastward forcing of the QBO phases. Furthermore, the Earth's QBO eastward phase is primarily induced by gravity waves, triggered by tropospheric convection (around 70% of the total eastward forcing) and Kelvin waves for the remaining 30% (Baldwin et al., 2001). To overcome the lack of eastward momentum in our Saturn stratospheric modeling, we plan to add a stochastic gravity wave drag parameterization (Lott et al., 2012) in our DYNAMICO-Saturn GCM. This is expected to produce a more realistic wave spectrum (with equivalent eastward-and westward-propagating waves), which would strongly impact the simulation of the equatorial oscillation and the downward propagation of winds.
Another main result of this study is the impact of ring shadowing on the stratospheric dynamics. In our DYNAMICO-Saturn reference simulation we obtained strong tropical eastward jets, which are seasonally periodic, and correlated with Saturn rings' shadow. With an additional simulation not including rings' shadowing, we show that the tropical eastward jets are caused by rings' shadowing in the stratosphere. Without rings' shadowing, the tropical eastward jets disappear after 3 simulated Saturn years. Comparisons between temperature predicted by our dynamical GCM simulations, computed with radiativeconvective equilibrium, and measured from Cassini/CIRS observations also suggests a dynamical impact of rings' shadowing in the stratosphere. The transformed Eulerian mean formalism shows that the dynamical impact of ring shad-owing on tropical eastward jets is strong both on the eddy-induced acceleration and the residual-mean-induced acceleration. In presence of rings' shadowing, eddy-induced acceleration is increased in the tropical channel and there is a reversal of the residual-mean-induced acceleration in the opposite hemisphere to the rings. Wave activity is correlated with rings' shadowing: the seasonality of the waves and eddies is enhanced by rings' shadowing, which contributes to the annual periodicity of the strong eastward tropical jets. Dynamics in the equatorial region, in particular the QBO-like oscillation, is also influenced by rings' shadowing.
Stratospheric winds have never been measured on Saturn and were determined using the thermal wind balance from the CIRS retrievals of stratospheric temperatures. Obviously, the tropical eastward jets could be a particularity of our DYNAMICO-Saturn GCM. Because of the small vertical wind shear associated with these tropical jets, there is no specific temperature signature associated to it. In other words, we cannot invalidate their existence with the available observations of the temperature field. Future measurements of winds in Saturn's stratosphere would help to validate the predictive scenario drawn by our model.
Future studies using the DYNAMICO-Saturn GCM could adopt a wider analysis scope and study the global meridional circulation in the stratosphere.
Results presented here are focused on the tropical regions, while the GCM extent is global. For instance, we could further characterize heat transport under the ring shadows, as well as the observed asymmetries of hydrocarbon abundances (Guerlet et al. (2009), Guerlet et al. (2010)). More generally, a complete study on the possible Brewer-Dobson-like circulation in the stratosphere of Saturn, and its impact on the hydrocarbons distribution, is warranted. A subsequent coupling of DYNAMICO-Saturn with photochemical models would then allow to refine this picture.
Sudden stratospheric warming on Earth's high and mid-latitude regions are associated with downward wind propagation anomalies of the Quasi-Biennial Oscillation (Lu et al., 2008). Is it possible that the extremely warm stratospheric disturbance in the aftermath of the Great White Storm of 2010-2011 (Fletcher et al., , 2012Fouchet et al., 2016) is due to a disruption of the downward propagation of the equatorial oscillation on Saturn? Conversely, Cassini observations showed that the occurrence of this huge storm disturbed the equatorial oscillations . Global stratospheric dynamical simulations by DYNAMICO-Saturn could help to address these questions in future studies.