Deep differential rotation and the inferred atmospheric metallicity of Jupiter and Saturn

Formation models of Jupiter [1] and seismologic constraints on Saturn [2] suggest that an extended fraction of their deep interiors may be inhomogeneous and stably stratified, while the outer envelope is largely convective. At present, it is not clear how efficiently angular momentum can be transported through a stable, dilute core. While the observed secular variation of the magnetic field of Jupiter constrains the differential rotation velocity at ~0.95 $R_{Jup}$, atop the primary dynamo region, to a small value of ~1 cm/s [3], here we assume that the stable deep interior may possess more significant differential rotation steadily enforced by cooling and contraction. This region contributes to the gravitational harmonics $J_2$ and $J_4$. Current Jupiter models predict atmospheric metallicites near or below the observed level of the equatorial water abundance [4]. Here, we apply a simple model of deep differential rotation to explore the relation between deep rotation rate and inferred atmospheric metallicity of Jupiter and Saturn through the coupling by $J_2$ and $J_4$. Our models aim to estimate the deviation from uniform rotation that would be required to enhance Jupiter's metallicity by +1x solar. The models are not meant to capture the complexity of the dynamics in their deep interiors. [1] Helled R., Stevenson D., ApJ 830:L4 (2017)[2] Mankovich, C., Fuller, J., eprint arXiv:2104.13385 (2021)[3] Moore K.M., Cao H., Bloxham J., Nat. Astron. 3:730 (2019)[4] Li C., Ingersoll A., Bolton S., et al., Nat. Astron. 4:609 (2020)