Zonal winds of Uranus and Neptune: Gravitational harmonics, dynamic self-gravity, shape, and rotation

Uranus and Neptune exhibit fast surface zonal winds that can reach up to few hundred meters per second. Previous studies on zonal gravitational harmonics and Ohmic dissipation constraints suggest that the wind speeds diminish rapidly in relatively shallow depths within the planets. Through a case-by-case comparison between the missing dynamical gravitational harmonic $J^\prime_4$ from structure models, and with that expected from fluid perturbations, we put constraints on zonal wind decay in Uranus and Neptune. To this end, we generate polytropic empirical structure models of Uranus and Neptune using $4^{\rm th}$-order Theory of Figures (ToF) that leave hydrostatic $J_4$ as an open parameter. Allotting the missing dynamical contribution to density perturbations caused by zonal winds (and their dynamic self-gravity), we find that the maximum scale height of zonal winds are $\sim 2-3\%$ of the planetary radii for both planets. Allowing the models to have $J_2$ solutions in the $\pm 5 \times 10^{-6}$ range around the observed value has similar implications. The effect of self-gravity on $J^\prime_4$ is roughly a factor of ten lower than that of zonal winds, as expected. The decay scale heights are virtually insensitive to the proposed modifications to the bulk rotation periods of Uranus and Neptune in the literature. Additionally, we find that the dynamical density perturbations due to zonal winds have a measurable impact on the shape of the planet, and could potentially be used to infer wind decay and bulk rotation period via future observations.