Physical origin and extent of Jupiter's extended shallow stably stratified layer, and the robustness of the Lowes-radius dynamo inference that implies it

Jupiter's magnetic spectrum, analyzed with the Lowes method, places the outer edge of dynamo action near 0.81-0.83 Jupiter radii - far deeper than the roughly 0.9 radii level below which the interior becomes strongly electrically conducting. Numerical dynamo simulations by Wulff, Cao and Aurnou (2025) suggest this can be reconciled only if Jupiter hosts an extended stably stratified layer running from about 0.8 up to at least about 0.9 Jupiter radii. But the top of such a layer lies deeper than where helium-hydrogen immiscibility ('helium rain') is expected to begin, so the layer's physical origin is unexplained, and the inference itself rests on untested assumptions. The open problem is to identify the origin, thermodynamics, and radial extent of this extended stable layer, and to validate the inference chain behind it: whether the flat-spectrum White Source Hypothesis holds at realistic dynamo control parameters, how non-flat spectra bias the inferred radius, whether the shallow secondary dynamo seen above deep stable layers causally requires a deep dynamo, and how zonal flows modify the magnetic filtering on which the method relies.

Background

Jupiter, like Earth, generates its magnetic field by dynamo action: the motion of electrically conducting fluid (liquid metallic hydrogen in Jupiter's deep interior) self-sustains the field through magnetic induction, a process governed by the magnetohydrodynamic (MHD) equations. Two structural features make Jupiter's dynamo harder to locate than Earth's. First, electrical conductivity does not jump at a sharp core-mantle boundary but rises smoothly with depth as hydrogen passes from a molecular, weakly conducting state to a metallic one; dynamically significant conductivity is reached only below about $0.9R_J$, where $R_J=71{,}492$ km is Jupiter's 1-bar equatorial radius. Second, interior structure models based on gravity measurements from NASA's Juno orbiter suggest that helium becomes immiscible in metallic hydrogen in the outer part of the conducting region. The resulting helium rain - helium droplets settling through hydrogen - imposes a compositional gradient that inhibits overturning convection, producing a stably stratified layer whose location and thickness are poorly constrained: one family of interior models prefers a thick stable layer between about 0.70 and 0.87 $R_J$ (Militzer and Hubbard 2024), while another argues for a layer as thin as a few tens of kilometres (Markham and Guillot 2024).

The dynamo radius is the outermost radial extent of dynamo action, and for a giant planet it can be estimated from the measured surface magnetic field by the Lowes method. The Lowes-Mauersberger spectrum $R_\ell$ is the magnetic energy per spherical harmonic degree $\ell$ (the surface field is expanded in spherical harmonics with coefficients fit to spacecraft measurements; Juno's reference model JRM33 has coefficients well determined through $\ell=13$ and useful field information through $\ell=18$; Connerney et al. 2022). If the region above the dynamo carries no electrical currents, the field there is a potential field and each $R_\ell$ decays outward as $r^{-2\ell-4}$; the radius at which the downward-continued spectrum becomes flat is taken as the top of the dynamo region. The flatness assumption - that the spectrum at the dynamo surface is white, i.e. independent of $\ell$ - is called the White Source Hypothesis. Applied to Jupiter, the method gives a Lowes radius of about $0.807\pm0.006\,R_J$ from JRM33 (Connerney et al. 2022) and $0.830\pm0.022\,R_J$ from an independent high-resolution field model (Sharan et al. 2022) - puzzlingly deep, given that conductivity becomes significant only near $0.9R_J$. Tsang and Jones (2020) showed that fully convective numerical Jupiter dynamos yield Lowes radii of 0.86-0.88 $R_J$, tied directly to the conductivity profile, and suggested that a stable layer in the outer envelope explains the discrepancy. Earlier dynamo simulations did include helium-rain-like stable layers (Gastine and Wicht 2021, whose layer lay at 0.84-0.88 $R_J$; Moore et al. 2022, whose favored layer lay at 0.9-0.95 $R_J$) but did not compute the Lowes radius of their models; independent zonal-wind analyses suggest additional stable stratification near $0.97R_J$, where Jupiter's deep winds are quenched (Christensen and Wulff 2024).

Wulff, Cao and Aurnou (2025) ran three-dimensional spherical-shell dynamo simulations with a Jupiter-like conductivity profile and a helium-rain stable layer at varying depths (all at Prandtl and magnetic Prandtl numbers $Pr=Pm=0.5$, Ekman number $E=3\times10^{-5}$, and Rayleigh number $Ra=1.2\times10^{8}$; the Prandtl and magnetic Prandtl numbers compare viscous diffusion to thermal and to magnetic diffusion respectively, the Ekman number measures viscous relative to Coriolis forces, and the Rayleigh number the convective driving). For shallow stable layers (upper boundary near $0.887R_J$) no significant electrical currents exist above the layer, the surface field is potential down to the layer base, and the inferred Lowes radius equals that base (about 0.80-0.82 $R_J$). For deeper layers, a shallow secondary dynamo operates in the conducting shell above the layer, and the inferred Lowes radius moves out to about $0.87R_J$, incompatible with Jupiter's observed spectrum. Matching Jupiter therefore requires a stable layer extending from about 0.81-0.83 $R_J$ up to at least about $0.9R_J$. The authors call this the extended shallow stable layer, and its interpretation is problematic: its upper boundary appears to lie deeper than the depth at which helium rain is expected to begin, so the layer cannot simply be the helium immiscibility region, and no alternative origin has been established. Because this conclusion flows from the Lowes method, its untested assumptions matter as well: the simulations show that the dynamo-region spectrum is not exactly flat, and the effect of a spectral exponent $\gamma\neq0$ (i.e. $R_\ell\propto\ell^{-\gamma}$, as in Kolmogorov-type turbulent spectra) on the inferred radius has not been quantified for gas giants. Zonal flows (longitude-averaged azimuthal winds) and shear within or above the stable layer further modify how the magnetic field is filtered and axisymmetrized on its way to the surface, an effect the present simulations cannot resolve in planetary regimes.

Problem Statement

Elucidate the physical origin, thermodynamics, and radial extent of the extended shallow stably stratified layer that the Lowes-radius comparison implies for Jupiter - a layer extending from roughly $0.81$-$0.83\,R_J$ up to at least about $0.9\,R_J$ - and establish whether the inference chain that produces this claim is valid at planetary conditions. Concretely, a resolution should:

1. Identify the mechanism that maintains a stable layer whose upper boundary lies deeper than the expected helium-rain initiation depth, and quantify its connection to helium immiscibility or to alternative compositional or thermal gradients, in a model consistent with Juno gravity and magnetic constraints (for example thermal-evolution, immiscibility, or layer-formation calculations that predict the layer's radial extent).

2. Determine, across realistic ranges of the MHD control parameters (Rayleigh number, Ekman number, Prandtl and magnetic Prandtl numbers), whether the magnetic spectrum at the top of the dynamo region satisfies the White Source Hypothesis ($\gamma=0$) or instead obeys $R_\ell\propto\ell^{-\gamma}$ with $\gamma\neq0$, and quantify the bias that a nonzero $\gamma$ introduces into the Lowes-inferred dynamo radius.

3. Establish whether the shallow secondary dynamo found above deep stable layers causally requires the presence of a strong deep dynamo - as opposed to independent coexistence - and identify the parameter thresholds, such as the local magnetic Reynolds number $R_m=u'_{rms}d_{layer}/\lambda$ built from the non-axisymmetric flow speed, the local layer thickness, and the magnetic diffusivity, that control its operation.

4. Quantify how zonal flows, differential rotation, and shear within or adjacent to the stable layer modify the filtering, axisymmetrization, and induction processes on which the Lowes inference relies, in regimes closer to planetary values than current simulations.

These deliverables constitute one scientific objective rather than a list of independent tasks: the existence, depth, and origin of the layer are claimed by way of the Lowes method, so the physical identification of the layer (item 1) and the validation of the method's assumptions at planetary parameters (items 2-4) stand or fall together. An answer that explains the layer while leaving the method untested, or that validates the method without addressing the layer's origin, does not close the problem. Conversely, the claim is refuted, and the problem is closed in the negative direction, if item 2 or item 4 establishes that the deep Lowes-inferred radius is an artifact of the method rather than evidence for an extended layer - specifically, if the spectrum at the top of the dynamo violates the flatness assumption through a nonzero spectral exponent ($\gamma\neq0$), or if zonal-flow filtering biases the inferred radius, so that no extended stable layer is required. Either one of these two refutations suffices on its own: the inference chain is then established as invalid, the layer that items 1 and 3 presuppose is removed, and items 1 and 3 are no longer required. Extension to parameter regimes and stable-layer thicknesses beyond the reach of present simulations (for example the tens-of-kilometres thin-layer scenario) enters only insofar as it is needed to discharge items 1-4.

References

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