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We present a review of Saturn's interior structure and thermal evolution, with a particular focus on work in the past 5 years. Data from the Cassini mission, including a precise determination of the gravity field from the Grand Finale orbits, and the still ongoing identification of ring wave features in Saturn's C-ring tied to seismic modes in the planet, have led to dramatic advances in our understanding of Saturn's structure. Models that match the gravity field suggest that differential rotation, as seen in the visible atmosphere, extends down to at least a depth of 10,000 km (1/6$^{\rm th}$ the planet's radius). At greater depths, a variety of different investigations all now point to a deep Saturn rotation rate of 10 hours and 33 minutes. There is very compelling evidence for a central heavy element concentration (``core''), that in most recent models is 12-20 Earth masses. Ring seismology strongly suggests that the core is not entirely compact, but is dilute (mixed in with the overlying H/He), and has a substantial radial extent, perhaps out to around one-half of the planet's radius. A wide range of thermal evolution scenarios can match the planet's current luminosity, with progress on better quantifying the helium rain scenario hampered by Saturn's poorly known atmospheric helium abundance. We discuss the relevance of magnetic field data on understanding the planet's current interior structure. We point towards additional future work that combines seismology and gravity within a framework that includes differential rotation, and the utility of a Saturn entry probe., Comment: Invited review. Accepted for publication in "Saturn: The Grand Finale", K. H. Baines et al., eds., Cambridge University Press. All-new follow-up to previous 2016 (pre-Grand Finale) review chapter here: arXiv:1609.06324

Interior modeling of Jupiter and Saturn has advanced to a state where thousands of models are generated that cover the uncertainty space of many parameters. This approach demands a fast method of computing their gravity field and shape. Moreover, the Cassini mission at Saturn and the ongoing Juno mission delivered gravitational harmonics up to J12. Here, we report the expansion of the Theory of Figures, which is a fast method for gravity field and shape computation, to the 7th-order (ToF7), which allows for computation of up to J14. We apply three different codes to compare the accuracy using polytropic models. We apply ToF7 to Jupiter and Saturn interior models in conjunction with CMS-19 H/He-EOS. For Jupiter, we find that J6 is best matched by a transition from He-depleted to He-enriched envelope at 2-2.5 Mbar. However, the atmospheric metallicity reaches 1xtimes solar only if the adiabat is perturbed toward lower densities, or if the surface temperature is enhanced by ~14 K from the Galileo value. Our Saturn models imply a largely homogeneous-in-Z envelope at 1.5-4x solar atop a small core. Perturbing the adiabat yields metallicity profiles with extended, heavy-element enriched deep interior (diffuse core) out to 0.4 RSat, as for Jupiter. Classical models with compact, dilute, or no core are possible as long as the deep interior is enriched in heavy-elements. Including a thermal wind fitted to the observed wind speeds, representative Jupiter and Saturn models are consistent with all observed Jn values., Comment: accepted to the Planetary Science Journal

Saturn's rings act as a system of innumerable test particles that are remarkably sensitive to periodic disturbances in the planet's gravitational field. We identify 15 additional density and bending waves in Saturn's C ring driven by the planet's internal normal mode oscillations. Taking advantage of a highly accurate absolute radius scale for the rings, we are able to detect weak, high-wavenumber (up to m=14) waves with km-scale radial wavelengths. From a systematic scan of the entire C ring, we report the discovery and identification of 11 new Outer Lindblad Resonances (OLRs), two counterpart inner Lindblad resonances (ILRs), and two new Outer Vertical Resonances (OVRs). The close agreement of the observed resonance locations and wave rotation rates with the predictions of models of Saturn's interior suggests that all of the new waves are driven by Saturnian f-mode oscillations. As classified by their spherical harmonic shapes, the modes in question range in azimuthal wavenumber from m=8 to 14, with associated resonance orders l-m ranging from 0 to 8, where l is the overall angular wavenumber of the mode. Our suite of detections for l-m=4 is now complete from m=8 to m=14. Curiously, detections with l-m=2 are less common. These newly-identified non-sectoral waves sample latitudinal as well as radial structure within the planet and may thus provide valuable constraints on Saturn's differential rotation. Allowing for the fact that the two ILR-type waves appear to be due to the same normal modes as two of the OLR-type waves, the 13 additional modes identified here bring to 34 the number of distinct f-modes suitable for constraining interior models., Comment: Accepted for publication in Icarus

We present Direct Numerical Simulations of the transport of heat and heavy elements across a double-diffusive interface or a double-diffusive staircase, in conditions that are close to those one may expect to find near the boundary between the heavy-element rich core and the hydrogen-helium envelope of giant planets such as Jupiter. We find that the non-dimensional ratio of the buoyancy flux associated with heavy element transport to the buoyancy flux associated with heat transport lies roughly between 0.5 and 1, which is much larger than previous estimates derived by analogy with geophysical double-diffusive convection. Using these results in combination with a core-erosion model proposed by Guillot et al. (2004), we find that the entire core of Jupiter would be eroded within less than 1Myr assuming that the core-envelope boundary is composed of a single interface. We also propose an alternative model that is more appropriate in the presence of a well-established double-diffusive staircase, and find that in this limit a large fraction of the core could be preserved. These findings are interesting in the context of Juno's recent results, but call for further modeling efforts to better understand the process of core erosion from first principles., Comment: Accepted for publication in ApJ

Fingering convection (or thermohaline convection) is a weak yet important kind of mixing that occurs in stably-stratified stellar radiation zones in the presence of an inverse mean-molecular-weight gradient. Brown et al. (2013) recently proposed a new model for mixing by fingering convection, which contains no free parameter, and was found to fit the results of direct numerical simulations in almost all cases. Notably, however, they found that mixing was substantially enhanced above their predicted values in the few cases where large-scale gravity waves, followed by thermo-compositional layering, grew spontaneously from the fingering convection. This effect is well-known in the oceanographic context, and is attributed to the excitation of the so-called "collective instability". In this work, we build on the results of Brown et al. (2013) and of Traxler et al. (2011b) to determine the conditions under which the collective instability may be expected. We find that it is only relevant in stellar regions which have a relatively large Prandtl number (the ratio of the kinematic viscosity to the thermal diffusivity), $O(10^{-3})$ or larger. This implies that the collective instability cannot occur in main sequence stars, where the Prandtl number is always much smaller than this (except in the outer layers of surface convection zones where fingering is irrelevant anyway). It could in principle be excited in regions of high electron degeneracy, during He core flash, or in the interiors of white dwarfs. We discuss the implications of our findings for these objects, both from a theoretical and from an observational point of view., Comment: To be published in ApJ

Jupiter's atmosphere has been observed to be depleted in helium (Yatm~0.24), suggesting active helium sedimentation in the interior. This is accounted for in standard Jupiter structure and evolution models through the assumption of an outer, He-depleted envelope that is separated from the He-enriched deep interior by a sharp boundary. Here we aim to develop a model for Jupiter's inhomogeneous thermal evolution that relies on a more self-consistent description of the internal profiles of He abundance, temperature, and heat flux. We make use of recent numerical simulations on H/He demixing, and on layered (LDD) and oscillatory (ODD) double diffusive convection, and assume an idealized planet model composed of a H/He envelope and a massive core. A general framework for the construction of interior models with He rain is described. Despite, or perhaps because of, our simplifications made we find that self-consistent models are rare. For instance, no model for ODD convection is found. We modify the H/He phase diagram of Lorenzen et al. to reproduce Jupiter's atmospheric helium abundance and examine evolution models as a function of the LDD layer height, from those that prolong Jupiter's cooling time to those that actually shorten it. Resulting models that meet the luminosity constraint have layer heights of about 0.1-1 km, corresponding to ~10,-20,000 layers in the rain zone between ~1 and 3-4.5 Mbars. Present limitations and directions for future work are discussed, such as the formation and sinking of He droplets., Comment: accepted to MNRAS, 21 pages, 17 figures