Dynamical constraints on the vertical structure of Jupiter's polar cyclones

TL;DR

The paper investigates how the vertical depth of Jupiter's polar cyclones affects their mean westward drift through the planetary $\\beta$-effect, i.e., $\\beta$-drift. It uses a 2D quasi-geostrophic model to quantify the drift as a function of the deformation radius $L_d$, and applies Bayesian nested sampling (Ultranest) to infer $L_d$ and cyclone parameters $(R,V,b)$ from five-year north- and south-pole drifts, yielding $L_d \\approx 226$ km (north) and $L_d \\approx 364$ km (south). To connect to 3D dynamics, the authors solve a vertical eigenvalue problem and obtain mode-dependent deformation radii $L_{d,(n)}=\\Gamma_n^{-1/2}$ with vertical profiles $\\Phi_n(z)$, illustrating how $N$ (the Brunt–Väisälä frequency) and depth $H$ determine plausible vertical structures. The framework links deformation-radius constraints to observable north-pole MWR footprints, enabling interpretation of vertical stability and providing a path to understand formation and persistence of Jupiter's polar cyclones.

Abstract

Jupiter's poles feature striking polygons of cyclones that drift westward over time, a motion governed by beta-drift (vortex motion caused by the latitudinal variation of the Coriolis force). This study investigates how beta-drift and the resulting westward motion depend on the depth of these cyclones. Counterintuitively, shallower cyclones drift more slowly, a consequence of stronger vortex stretching. By employing a 2D quasi-geostrophic model of Jupiter's polar regions, we constrain the cyclones' deformation radius, a key parameter that serves as a proxy for their vertical extent, required to replicate the observed westward drift. We then explore possible vertical structures and the static stability of the poles by solving the eigenvalue problem that links the 2D model to a 3D framework, matching the constrained deformation radius. These findings provide a foundation for interpreting upcoming Juno microwave measurements of Jupiter's north pole, offering insights into the static stability and vertical structure of the polar cyclones. Thus, by leveraging long-term motion as a novel constraint on vertical dynamics, this work sets the stage for advancing our understanding of the formation and evolution of Jupiter's enigmatic polar cyclones.

Figures (9)

• Figure 1: Impact of Stretching on $\beta$-Drift Dynamics in a Single-Layer QG Framework: A Single Cyclone on a $\beta$-Plane. (a--b) Evolution of the zonal ($u_\beta$) and meridional ($v_\beta$) drift velocities over time for four different values of the Burger number and $b=1.2$. Dashed curves show exponential fits with amplitude $\rm{U}_\beta$ and timescale $\tau$. Here, $(u_\beta,v_\beta)$ are non-dimensional, scaled by $\hat{\beta}V$. (c) Plots of $\rm{U}_\beta$ (solid line, left axis) and $\tau$ (dashed line, right axis, in rotation periods) as functions of ${\rm Bu}$ for three values of the shape factor $b$ (legend in panel d). The exponential parameterization becomes inadequate for ${\rm Bu} \gtrsim 5$ (see Fig. S2). (d) Solid line: Phase angle $\alpha$ (defined in panel e) between $u_\beta$ and $v_\beta$ as a function of ${\rm Bu}$. Dashed line: Strength of the $\beta$-gyres, quantified as $\Delta\psi_g=\max(\psi_g)-\min(\psi_g)$, with values non-dimensionalized by $\hat{\beta}RV$. (e) The vorticity field ($\xi_{\rm g}$, in color), streamfunction ($\psi_{\rm g}$, in contours), and velocity field ($\mathbf{u}_{\rm g}$, arrows) after five rotation periods for $\rm{Bu}=0.59$ and $b=1.2$. The $\beta$-gyres appear as red and blue lobes surrounding the cyclone core. The colorbar represents the scaled (by $\hat{\beta}V/R$) vorticity field. See Movie S1 for the simulation.
• Figure 2: Constraining $L_d$ at Jupiter's Poles Using Observed and Simulated Cyclone Drifts. (a,b) Observed trajectories of the north (a) and south (b) polar cyclones over five years mura2022five_years. The symbols $\odot$ and $\otimes$ mark the start and end points, respectively. The background is a JIRAM infrared image from PJ4 (Image from adriani2018), corresponding to the onset of these trajectories. (c,d) Simulated trajectories after five years for a model that reproduces the observed mean westward drift. The color scale indicates the magnitude of the velocity vector, while the streaks represent the flow direction through line integral convolution (LIC) of the velocity field at $t=0$. (e--j) Sample model trajectories from the nested sampling routine, illustrating diverse outcomes. The numbers above the panels are the mean and standard deviation (between the cyclones) of the westward drift. The parameters used in each panel are listed below the panels. (k--n) Histogram plots of the posterior parameter distributions produced by Ultranest buchner2021ultranest, after sampling $15{,}000$ parameter sets per pole for the north (blue) and south (orange) polar cyclones. The dashed lines denote the median, and the dotted lines mark $\pm1\sigma$ for the distributions. These values are written above the panels. Markers represent the parameter values of the successful models (panels c,d). For a corner plot of the joint posterior distributions, see Fig. S4.
• Figure 3: Interpretation of the Cyclones' Vertical Structure from the Deformation Radius. The first row corresponds to the "mode 0" solutions of the eigenvalue problem described by Eq. \ref{['eq: eigenvalue problem, depth']}, while the second row corresponds to "mode 1". (a,d) Vertical profiles of $\Phi_n$ for each mode. (b,e) Solutions for $L_d$ (green shades) for various combinations of $N$ (Brunt-Väisälä frequency) and $H$ (depth). The estimated $L_d$ distributions from Fig. \ref{['fig: 2D polar model fig depth']}k are shown by the blue and orange curves for the north and south poles, respectively. (c,f) Solutions for $\Phi_n$ as a function of pressure below the cloud layer, following the $N$ and $H$ relationship indicated by the blue solid curve in panels (b) and (e). Green horizontal dotted lines indicate the depths at which each MWR channel has maximum sensitivity Janssen_2017, though each channel is sensitive over a broader range of depths. The numbers on the right ordinate show the altitude relative to 1 bar. Grey dashed lines in panels (b,c,e,f) denote the values used to generate the profiles in panels (a) and (d), and to produce Fig. \ref{['fig: 3D fig']}. The color along the grey dashed line in panels (c,f) corresponds to the $\Phi(z)$ profiles shown in panels (a,d). The purple dashed line in panel (c) represents the theoretical scaling $H \sim \mathrm{Ro}^{1/2} R f_0/N$ (adapted from Aubert2012), using representative values of $R=800 \,\rm{km}$ and $U=90\,\rm{ms}^{-1}$ from Fig. \ref{['fig: 2D polar model fig depth']}m-n.
• Figure 4: Vertical Structure and Intersections with Instrument Sensitivity Depths at the North Pole. The $\psi$ field from Fig. \ref{['fig: 2D polar model fig depth']}c was extended downward (rather than computed directly in a 3D simulation) using the eigenfunctions from Fig. \ref{['fig: Depth eigenprob']}a,d, which correspond to $N = 3 \times 10^{-3}\;(\mathrm{s}^{-1})$ and $H = 50$ ($133$) km for mode 0 (mode 1). Both modes yield $L_d = 226$ km, with mode $0$ terminating ($\psi=0$) at $z=-50$ km, and mode 1 continuing further to $\psi=0$ at $z=-134$ km. The planes represent $\psi$ at the MWR channels that overlap with the eigenfunction range, while the shaded regions between the planes depict $\psi$ values between the channels. The dashed black line indicates the north pole. The background image on channel 6 is a JIRAM measurement from PJ4 adriani2018, similar to Fig. \ref{['fig: 2D polar model fig depth']}a.
• Figure S1: Illustration of how stretching affects shallow-water flow. (a) A tranquil, rotating fluid layer with a solid lower boundary, a free surface, and mean depth $H$. (b) A significant perturbation induces strong horizontal convergence, producing a depth anomaly $\delta h$ comparable to $H$ and resulting in a large increase in relative vorticity. (c) Similar convergence occurs in a much deeper layer ($H \gg \delta h$), leading to a negligible relative vorticity anomaly due to the weaker effect of stretching.