Model of deep zonal flows in giant planets

TL;DR

This work demonstrates that a thin, stably stratified layer near a planet's surface, driven by a localized forcing, can generate a meridional circulation that establishes a horizontal temperature gradient. Through 3D rotating convection simulations and a complementary basic linear/asymptotic analysis, the authors show that, when the horizontal temperature anomaly persists into the interior, thermal wind balance yields jets that decay with depth in a Jupiter-like manner; conversely, strong forcing or weak convection tends to confine the jets to boundary layers. The study identifies a regime where fluctuations are dominated by convection versus those driven by the imposed shear, with deeper penetration favored in convection-driven cases. These findings offer a framework for interpreting Jupiter’s gravity data and motivate further enhancements including MHD effects, compressible/anelastic dynamics, and spherical geometry. Key result: the depth profile of zonal winds is controlled by the persistence of the top-layer horizontal temperature anomaly under thermal wind balance, modulated by the interplay between forcing and convection.

Abstract

A mechanism by which the surface zonal flows of giant planets can be gradually attenuated with depth is explored. The zonal flow is driven by an imposed forcing in a thin layer near the surface. A meridional circulation is set up, analogous to the Ferrel-like cells observed in Jupiter's atmosphere. Acting on a stably stratified thin surface layer, the meridional flow induces a horizontal temperature anomaly which leads to a gradual reduction of the zonal winds with depth, governed by the thermal wind equation. Our model is a Boussinesq plane layer, with gravity acting parallel to the rotation axis. A suite of fully three-dimensional time-dependent numerical simulations has been performed to investigate the model behaviour. Below the forced stable layer, convection is occurring, typically in the form of tall thin cells. The fluctuating components of the three-dimensional flow can be driven by either the convection or the Reynolds stresses associated with the jet shear flow. When fluctuations are mainly driven by convection in the form of tall thin columns and the forcing is relatively weak, the horizontal temperature anomaly persists much deeper into the interior than when it is driven by shear flow. The zonal jets can therefore extend deep into the interior, consistent with the Juno gravity data.

Figures (11)

• Figure 1: (a) The deep zonal flow model of Kaspi et al. Kaspi23. The $z$-coordinate is parallel to the rotation axis, and the equivalent latitude at the surface is shown (for details see Appendix A). Positive wind speeds correspond to eastward flow in metres/sec and negative wind speeds correspond to westward flow. Only latitudes between $\pm 30^\circ$ are shown. The dashed line is at a depth of 3500 km below the surface. (b) A zonal flow profile consistent with the gravity data, for the jet at latitude $21^\circ$ N. The $z$-coordinate here is scaled so that $z=0$ is the bottom of the layer, $z=1$ is the top of the layer, see Appendix A for details. The unscaled layer depth is 12780 km.
• Figure 2: Simulation results for run A8 with no stable layer (Q=0) and run A9 with a strong stable layer (Q=200). (a) shows the basic state temperature as given by equation (\ref{['eq:basicstate']}) for the two runs. (b) and (c) show the mean zonal flow, $\bar{u}$, as a function of $z$ at $y=0$. (d) and (e) show the mean temperature perturbation $\bar{\theta}$. Note that in the two runs the sign of $\bar{\theta}$ is reversed in the $y$-direction. Panels (f) and (g) show the $x$-averaged vertical velocity at a snapshot in time, $\langle w \rangle_x$. These are broadly similar, but the columnar structure of the convection is particularly visible in run A9 (g).
• Figure 3: Approximate thermal wind balance for run A9. The two terms in equation (\ref{['thermal_wind_eq']}) are shown to be in fairly good agreement, so that the inertial and viscous terms in equation (\ref{['full_thermal_wind_eq']}) are negligible in this case, and $\epsilon_{TW} = 0.098$ only.
• Figure 4: Further results from run A9. (a) shows a snapshot of the vertical velocity at the horizontal mid-plane, $z=0.5$. (b) shows a snapshot of the vertical velocity in the $y-z$ plane at $x=0$. The columnar nature of the flow is evident here. (c) the mean zonal flow, $\bar{u}$, consistent with Fig. \ref{['figs:fig3']}(a). (d) the $y$-component of the mean flow, $\bar{v}$ which is much stronger in the boundary layers than in the interior. (e) The mean temperature perturbation, $\bar{\theta}$ in the region $0.2 < z < 0.8$ (see Fig. \ref{['figs:fig2']}e for the full version) showing how the temperature perturbation decays with depth. ${\bar{\theta}}_{vr}=0.27$ for this run. (f) The full mean temperature, $T_{BS}+\bar{\theta}$, in the region $0.2 < z < 0.8$.
• Figure 5: Comparison of the basic linear state model with simulations from run A14. (a) shows the mean zonal flow $\bar{u}(y,z)$. (b) shows the mean temperature perturbation $\bar{\theta}$. Profiles from the nonlinear simulation (blue solid line), basic linear state model (orange dashed line) and asymptotic theory of the basic linear state model (black dotted line) are shown for $\bar{u}$ at $y=0$, $\bar{v}$ at $y=0$, $\bar{w}$ at $y=1$ and $\bar{\theta}$ at $y=1$ in (c), (d), (e) and (f), respectively.