Jones (2014) obtained self-consistent, dipole-dominated, Jupiter-like dynamo solutions with an anelastic convection-driven model based on an ab initio reference state, but found them only in a narrow window at low Prandtl number (Pr≈0.1Pr\approx0.1) and Ekman numbers near 10−510^{-5}, with the computational domain truncated 3000 km below the cloud tops and the diffusivities calibrated to a then-tentative secular variation of Jupiter's field. The work explicitly left open whether such dipolar solutions occupy a substantially larger region of parameter space at planetary Ekman numbers, how sensitive they are to the domain cut-off radius and the electrical-conductivity transition profile, whether the turbulent-diffusivity calibration (342 m^2/s, about 700 times the laminar value) is correct, and whether deep zonal flows can coexist with a dipole-dominated dynamo — together, whether these models can be validly extrapolated to Jupiter, whose Ekman number (∼10−15\sim10^{-15}) and laminar magnetic Reynolds number (∼105\sim10^5) are numerically unreachable. Juno-era work (2017–2026) resolved the observational premises (Jupiter's secular variation is now firmly established; the zonal winds are measured to decay near 0.95 Jupiter radii) and demonstrated jet–dynamo coexistence at E=10−6E=10^{-6} with stable stratification, but the dipolar window below E=10−5E=10^{-5} remains unmapped and the diffusivity-scaling question remains explicitly open in the current literature.

Background

Jupiter, like Earth, generates its magnetic field by a hydromagnetic dynamo: convective motion of electrically conducting fluid — liquid metallic hydrogen in Jupiter's deep interior — sustains the field against ohmic decay through electromagnetic induction. Because Jupiter's interior is strongly density-stratified (density varies by orders of magnitude between surface and center), modelers use the anelastic approximation, a filtered form of the compressible magnetohydrodynamic equations that retains density stratification but removes sound waves, rather than the incompressible (Boussinesq) equations used for Earth's core. The radial profiles of density, temperature, and electrical conductivity come from a reference state derived from ab initio (first-principles quantum) simulations along Jupiter's adiabat (French et al. 2012). In particular, electrical conductivity rises smoothly but very steeply from negligible values in the molecular-hydrogen envelope to metallic values below roughly 0.9 Jupiter radii, with no sharp boundary; this gradual transition is central to the questions below. The standard control parameters are the Rayleigh number (buoyant driving relative to viscous and thermal diffusion), the Ekman number EE (viscous relative to Coriolis force; Jupiter's is of order 10−1510^{-15}), the Prandtl number PrPr (momentum diffusivity relative to thermal diffusivity), and the magnetic Prandtl number PmPm (momentum relative to magnetic diffusivity); the magnetic Reynolds number RmRm (advection relative to magnetic diffusion) must exceed order 10–100 for dynamo action. Prior numerical work had shown that dipole-dominated solutions (fields dominated by the axial dipole rather than smaller-scale multipolar harmonics) are hard to obtain in anelastic gas-giant models: strong background density stratification biases dynamos toward small-scale multipolar fields (Gastine et al. 2012), and strong zonal (east–west) winds interacting with radially varying conductivity shear the field into non-dipolar configurations (Duarte et al. 2013; Gastine et al. 2014). Jones (2014) nevertheless found steady dipole-dominated, Jupiter-like solutions, but only in a narrow window at low Pr≈0.1Pr\approx0.1 with Pm>1Pm>1 and E≈1.5E\approx1.5–2.5×10−52.5\times10^{-5}, and with the model domain truncated at a cut-off radius 3000 km below the cloud tops, above which electrical conductivity is simply neglected. In these solutions a strong magnetic field suppresses differential rotation inside the conducting region; the strong surface zonal winds cannot penetrate the dynamo region, which the paper places about 7000 km deep. These solutions are far from Jupiter's actual parameters: EE is ten or more orders of magnitude too large, the diffusivities are effective 'turbulent' values orders of magnitude above molecular ones, and the laminar RmRm of order 10510^5 estimated for Jupiter is unreachable numerically. All extrapolation from simulation to planet therefore rests on scaling laws (Christensen and Aubert 2006) and on a calibration in which the model's effective turbulent magnetic diffusivity (342 m^2/s, about 700 times laminar) is fixed by matching a tentative observed secular variation — the slow time variation of the main field — of 0.012% per year in the axial dipole coefficient. The Juno spacecraft, in polar orbit around Jupiter since 2016, has since delivered high-precision magnetic-field, secular-variation, and gravity data, giving each of these open questions direct observational leverage.

Problem Statement

Jones (2014) established that self-consistent, dipole-dominated, Jupiter-like dynamo solutions exist in anelastic convection-driven models, but only in a narrow low-Prandtl-number window near E∼10−5E\sim10^{-5}, and posed the extrapolation of these solutions to Jupiter's actual parameters as the central open problem. The problem is to determine whether — and under what conditions — such dipole-dominated solutions characterize Jupiter's real parameter regime, by resolving four linked questions raised in that work. (1) Low-Ekman-number window: do dipole-dominated solutions occupy a substantially larger region of parameter space at Ekman numbers below 10−510^{-5}, with the lower molecular and turbulent diffusivities and more realistic turbulent Prandtl numbers this implies, or does the low-PrPr window found at E∼10−5E\sim10^{-5} close or shift as planetary values are approached? (2) Domain and conductivity sensitivity: how sensitive are the dipolar solutions to the domain cut-off radius — in particular, does reducing the cut-off depth below 3000 km, which would enhance turbulence in the outer semiconducting layer, feed back on and modify or destroy the dipolar dynamo — and to the detailed profile of the electrical-conductivity transition between the molecular and metallic regions? (3) Turbulent-diffusivity calibration: is the mapping from the dimensionless simulation diffusivities to planetary turbulent transport coefficients correct — specifically the calibration in which an effective turbulent magnetic diffusivity of 342 m^2/s (about 700 times the laminar value) is inferred from Jupiter's secular variation — and are the derived internal field magnitudes robust to alternative diffusivity scalings? (4) Coexistence with deep zonal flows: can deep zonal flows of realistic amplitude, which grow relative to the convective flows as EE decreases, coexist with a dipole-dominated dynamo under a realistic conductivity profile, or does the field inevitably either suppress the winds or lose its dipolar form to them? Because Jupiter's Ekman number (∼10−15\sim10^{-15}) and laminar magnetic Reynolds number (∼105\sim10^5) cannot be simulated directly, an answer must establish the trend at achievable lower EE together with a validated extrapolation argument — a scaling law, an asymptotic theory, or an observational calibration — covering the remaining gap. The four questions share one objective: determining whether Jones-type anelastic dynamo models can be validly extrapolated to real Jupiter, and each answer must state how its conclusions bear on that common objective. All four questions are mandatory and jointly required for a complete answer — resolving any subset, including the low-Ekman-number window (1) alone, does not by itself resolve the task — and within question (2) the 3000 km cut-off reduction is an illustrative instance of the cut-off-radius sensitivity rather than the sole required target.

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