A question about a patch

The Juno spacecraft, in polar orbit around Jupiter since 2016, produced the first sharp map of a giant planet’s magnetic field. The reference model distilled from its prime mission — JRM33 (Connerney et al. 2022), a degree-30 spherical-harmonic fit to 32 polar orbits — revealed a planet magnetically unlike Earth: nearly all the northern flux funnels through a narrow intense band, an isolated patch of inward flux (the Great Blue Spot) sits almost on the equator, the southern hemisphere is smooth and diffuse — the hemispheric dichotomy that Moore et al. (2018) read as evidence of a structurally complex dynamo — and, most relevant here, a region of reversed flux near the north pole.

A patch of reversed polarity embedded in a dominant cap is a configuration this series has met before. On the Sun it is called a parasitic polarity, and it generically comes with a complete topological skeleton: a magnetic null point floating above the patch, whose fan separatrix closes down around the parasite as a dome footprinted on the polarity-inversion line, and whose spine threads the dome — one foot inside the patch, the other connecting far away. So the question for Jupiter is precise: is the polar patch a combination of separatrices — dome plus spine, the solar anatomy on a giant planet? Nobody hands us the answer: separatrix analyses at Jupiter exist at magnetospheric scale (how the aurora maps to open and closed field lines — see Zhang et al. 2021), but the internal field’s near-surface skeleton is exactly what our toolkit measures.

Everything below runs on the real coefficients (via the open planetmagfields package), turned into a vector field by our own Schmidt-normalised evaluator — golden-checked against the package’s maps to $5\times10^{-16}$ relative error and against an analytic dipole exactly.

The field, and where the patch actually lives

The JRM33 model field (Juno-derived), at two depths. A: the radial field $B_r$ of JRM33 truncated at degree 18 (the model itself carries coefficients to degree 30; $l \le 18$ is the useful-information cut, $l \le 13$ the well-determined core — see the robustness section) at the 1-bar surface ($r = 1\,R_J$; red/blue = field out of/into the planet, colour scale in Gauss; black curve = the polarity-inversion line $B_r = 0$). The hemispheric dichotomy is plain — a structured north against a smooth south — and the Great Blue Spot sits at the equator near right-handed System III longitude 275° (≈ 85°W). Colour limits as rendered (each panel has its own colourbar): A ±18 G, B ±12 G, C ±22 G. B: the north polar cap at $r = 1$: one polarity, no reversed flux anywhere poleward of 45°N. C: the same cap at $r = 0.85\,R_J$ — JRM33's conventional mapping surface near the top of the dynamo region (the Lowes-radius estimate for the dynamo top is $\approx 0.81\,R_J$): patches of reversed (inward) flux appear, including the elongated polar patch at latitude 65–75°N — the subject of this post. Between these two depths lies Jupiter's molecular envelope: weakly conducting hydrogen, current-free to a good approximation — potential-field territory, the same mathematics as the solar corona of Parts 4–6. Pipeline: src/jupfield.py (golden-checked), scripts/render_j1_figures.py.

Panel B against panel C is the first finding, and it dictates where to hunt. At the visible surface the northern cap is unipolar — the reversed patch exists only at depth. Jupiter is a gas giant: $r = 1\,R_J$ is just the 1-bar pressure level, and the shell between the top of the dynamo region ($r = 0.85\,R_J$ is the conventional mapping surface; metallic hydrogen conducts and generates the field below roughly the Lowes radius $\approx 0.81\,R_J$) and the cloud deck is the planet’s “corona” — a region the field crosses as a nearly potential field. If the patch has a null and a dome, they must live inside the planet’s envelope, between 0.85 and 1 $R_J$. We amended the hunt accordingly and record the amendment: the first sweep, restricted to $r > 1$, found zero nulls in every model — the dipole simply dominates above the clouds.

The hunt: two nulls from the seeds — and a third from the census

Newton descent on the analytic mode sum, seeded densely through the shell ($\sim$3,000 seeds, both hemispheres), finds two magnetic nulls in the $l \le 18$ JRM33 continuation. The two-resolution cell-prefiltered root census the review demanded (run_j2_census.py: Haynes–Parnell-style corner-sign prefilter over every spherical cell, Newton descent from candidate-cell centres, two grid resolutions agreeing — convergence evidence, not a completeness theorem: the corner-sign filter is necessary-only and centre-started Newton can fail or leave its cell) then confirmed these two — and found a third the seeding had missed, sitting just above the shell floor in the equatorial band of reversed patches. A concrete lesson, banked: seeds are not a census — and this census is not a completeness proof either.

null	r [$R_J$]	latitude	longitude	type	degree	$J_\parallel$ (dimensionless)	$Q$
polar	0.873	+69.0°	282.3°	radial+	$+1$	$0.00$	6
low-latitude	0.864	+9.7°	157.4°	radial+	$+1$	$0.00$	6
census find (J2)	0.857	+27.2°	67.0°	radial−	$-1$	$0.00$	6 (tangent-cone; raw $w_4$ not measured)

(The census find lies $0.002\,R_J$ above the census floor at $0.855$, so the floor was tested: rerunning the census with the shell floor lowered to $0.80$ (run_j2b_sensitivity.py) returns exactly the same three nulls in the $0.855$–$1.0$ shell — the find is not a boundary artifact — while exposing a root-dense layer below $\approx 0.85$ (39 further roots of the truncated continuation in $0.80$–$0.855$: a truncation-sensitive, poorly constrained continuation region — whether noise, downward-continuation amplification, or unresolved real structure dominates there is not diagnosed, and no completeness is claimed; that layer is why the census shell starts at $0.855$). Anatomy of the find: radial, topological degree $-1$ — an opposite-sign partner to the other two. And a topological cross-check brackets the shell from above: the degree of $\mathbf B/\lvert\mathbf B\rvert$ over spheres at $r = 0.885$, $0.94$ and $0.999$ is numerically consistent with zero ($-0.005$, $-0.000$, $-0.000$), so the census leaves no unexplained net topological charge above $0.885$ — index-cancelling missed pairs are not excluded (fold-created nulls come precisely in such pairs). The three found nulls all live in $0.855$–$0.885$, where the two-resolution and floor-move agreement is convergence evidence, not completeness; the stronger comparator, a Haynes–Parnell trilinear cell census, was not performed.)

The two seeded nulls sit a few hundredths of a radius above the dynamo surface — and both land exactly where the theory of Parts 4–5 says they must. The growth vector reads $Q = 6$ at each — and this time the reading has a non-circular leg. Because the extrapolation is analytic, its internal harmonics admit an exact closed-form vector potential ($A = -\tfrac1l r^{-(l+1)}\,\hat r\times\nabla_s S_{lm}$ per harmonic, golden-checked $\nabla\times A = \mathbf B$), and the flux-reach exponent can be measured on the full field itself, no Jacobian input anywhere: $w_4(r)$ reads $\approx 3$ across the whole radii ladder at both nulls ($2.7$–$3.1$) and $\approx 2$ at a generic control point (scripts/run_j1d_raw_w4.py) — the $k=1$ flux weight, read raw, on the fourth world. (The tangent-cone $Q = 6$ in the table remains what it is everywhere on modelled fields: a consistency check given the measured Jacobian.) All three nulls are radial with $J_\parallel = 0.00$ ($J_\parallel \equiv (\nabla\times\mathbf B)\cdot\hat{\mathbf s} /\max_i\lvert\lambda_i(\nabla\mathbf B)\rvert$ — the spine-parallel curl in units of the Jacobian’s largest eigenvalue modulus, dimensionless, as in Part 6): the envelope field is curl-free, so the force-free theorem of Part 5 forbids spiral nulls here — and the census obliges, 0 spirals out of 3.

The buried skeleton. The dynamo surface ($r = 0.85\,R_J$) textured with the real JRM33 $B_r$ (red out / blue in, colour scale saturating at ±20 G; the deep-blue region at lower left is the Great Blue Spot), with the polar null (gold ★, $r = 0.873\,R_J$), its fan separatrix dome closing onto the polar reversed patch, and its spine — the inner foot rooted in the patch, the outer branch arcing away (drawn fading; it lands in the far southern hemisphere at latitude $-53°$). The faint shell is the 1-bar cloud surface: the whole structure lives inside the planet, its ceiling roughly 9,000 km beneath the visible clouds. Occlusion-culled real field lines; no schematic elements.

The answer: dome plus spine

Tracing the fan surface of the polar null — 72 field lines launched around its fan plane — every single line closes down onto the dynamo surface, and their footprints form a closed curve. The question “is the patch a combination of separatrices?” now has a quantitative answer:

• The dome footprint traces the patch boundary. Median angular distance from a fan footprint to the nearest point of the patch’s polarity-inversion line: 2.3° (maximum 3.9°). In the other direction — from the boundary to the nearest footprint — the median is 5.3°, with a 90th percentile of 15° where the patch’s thin western extension outruns the 72-line discretisation of the dome.
• The spine completes the skeleton. One spine branch lands inside the patch (latitude 69°N, longitude 281° — the parasite’s umbilical); the other exits the dome, arcs through the envelope, and lands in the southern hemisphere at latitude $-53°$ — the polar patch is magnetically wired to the opposite side of the planet.
• The low-latitude null repeats the anatomy over its own small reversed patch (footprint-to-boundary median 1.5°): the configuration is not a polar accident but the generic response to parasitic flux.

So in the $l \le 18$ JRM33 continuation: yes — the polar patch is precisely a combination of separatrices. Its flux is enclosed by a fan dome whose footprint traces the patch boundary, pierced by a spine — the textbook parasitic-polarity skeleton of solar physics, transplanted to a giant planet and buried under the clouds. And that burial explains panel B of the first figure: the dome’s ceiling ($r = 0.873$) sits below the 1-bar level, so by the time the field reaches the visible surface the parasite has been fully covered — the cap looks unipolar because the separatrix dome closes the reversed flux before it can reach daylight.

The dome footprint against the patch boundary — the answer, drawn. $B_r$ at the dynamo surface (red/blue; dashed black = the $B_r = 0$ polarity-inversion lines). A: the north polar cap: the 72 fan-line footprints (orange) of the polar null (gold ★) trace the boundary of the reversed patch — median footprint-to-boundary distance 2.3°, maximum 3.9° — and the inner spine footpoint (green) roots inside it. B: the low-latitude null over its own small parasite (footprint-to-boundary median 1.5°) amid the equatorial band of reversed patches. Artifacts: artifacts/j1_jupiter.json, j1b_dome.json; pipeline scripts/run_j1_jupiter.py + run_j1b_dome.py.

The two seeded nulls, dissected (the third, census-found null is profiled in the text, not in this sheet). The standard null-frame anatomy sheet (as for the Sun and the magnetosphere in Part 6; colour = topological role, after Pontin & Priest 2022): a 3D skeleton with the translucent ideal fan disc, a view down the spine where the real fan lines (blue) radiate, and a side view with the spine (orange) vertical — where the polar null's fan visibly curves downward: that bending sheet is the dome of the skeleton figure closing onto the dynamo surface. Gray = ambient lines; open circle coloured by topological degree (blue $+1$, red $-1$); dashed circle/line = the ideal fan plane; axes in $R_J$, linear. A–C: the polar null ($r = 0.873\,R_J$, latitude $+69°$; $\nabla\mathbf B$ eigenvalues normalised by the largest $\lvert\lambda_i\rvert$, dimensionless: $(1.00, -0.27, -0.73)$). D–F: the low-latitude null ($r = 0.865\,R_J$, latitude $+10°$; eigenvalues $(1.00, -0.07, -0.93)$). Both are clean radial nulls of degree $+1$ with $J_\parallel = 0.00$ — the vacuum envelope permits nothing else (Part 5's theorem) — and the SR growth vector reads $Q = 6$ at both: the law's fourth-world consistency check.

A boundary can be a separatrix without a null

One alternative had to be ruled in or out before the dome could take all the credit. A polarity-inversion line can carry separatrix character with no null at all, through bald patches — PIL segments where the field grazes the surface, $(\mathbf B\cdot\nabla)B_r > 0$, so that field lines touch and skim outward instead of arching over (Titov, Priest & Démoulin 1993). If Jupiter’s polar PIL were bald-patch dominated, “the patch is a combination of separatrices” would be true for a reason that has nothing to do with the null we found. So we classified the actual PILs, point by point (scripts/run_j1c_baldpatch.py):

• The polar patch at degree 18: bald-patch fraction 0.000. No sampled point of its boundary is grazing-type — so whatever separatrix character the polar PIL has is carried by the null’s fan dome, with no bald-patch contribution. (Stated carefully: the zero fraction rules out the alternative mechanism; how much of the boundary the dome actually accounts for is bounded by the footprint distances above — median 2.3° one way, p90 15° the other — not by this test.)
• The low-latitude patch: fraction 0.234 — a mixed boundary, dome plus grazing segments, the common situation on the Sun.
• The degree-13 survivor (the weak polar patch that keeps existing after the null disappears): fraction 0.104. With the dome gone, only a tenth of its boundary retains separatrix character as bald-patch segments; the rest is plain arcade. So under truncation the answer genuinely degrades — the patch does not quietly stay separatrix-bounded by another mechanism.

The robustness protocol, applied to our own claim

This series does not publish a topology claim without trying to kill it. The spherical-harmonic analogue of Part 6’s window-shift test is model truncation and model exchange: JRM33’s coefficients are, in Connerney et al.’s own words, reasonably well determined through degree 13; degrees 14–18 carry the small-scale power that downward continuation to $r = 0.85$ amplifies by factors of $(1/0.85)^{l+2}$ — a factor of ~13 at $l = 14$. So:

• Truncated to degree 13: the polar reversed patch survives, but barely (minimum $B_r \approx -0.75$ G against $-6$ G at degree 18) — and the envelope holds no nulls at all: the weakened parasite no longer reverses the vertical field balance above it, so no dome forms.
• JRM09 (the 9-orbit predecessor, degree 10): no reversed flux at the polar patch location, and its own envelope nulls sit elsewhere (three near the equator, two in the south) — at that resolution the polar parasite is simply not in the model.

In this model, the dome-plus-spine skeleton is what the $l \le 18$ JRM33 continuation says lies beneath Jupiter’s north pole — the patch is a combination of separatrices in that modelled field — but the structure rides on the degree-14–18 coefficients, the least-determined tail of the model. The inferred structure depends on model resolution, as do the low windowed nulls in Part 6. For the uncertainty test, formal JRM33 covariances are not distributed with the coefficients, so the pipeline builds a stress ensemble — per-degree Gaussian perturbations scaled by the JRM33−JRM09 per-degree coefficient change (1.3% at $l = 2$, 23% at $l = 10$, log-linearly extrapolated to essentially unconstrained at $l \ge 14$). This is a deliberately constructed stress model — a proxy, honestly labelled, not a posterior, so its outputs are survival fractions under the chosen stress ensemble and sample position ranges, not posterior probabilities or formal uncertainties. Across $N = 40$ members (run_j2b_sensitivity.py, all three nulls tracked): the polar reversed patch survives in 40/40 — at every tested threshold ($0.25$, $0.5$, $1$ G) — and the polar null in 32/40 (acceptance: the first root within $0.15\,R_J$ of the nominal null, then the shell-floor cut $r \ge 0.856$; an earlier census run that applied the floor inside the root-finder counted 33/40 with longitudes from $262°$, because in one member a different root stood in for the polar one, which had sunk to $r = 0.81$ — run_j2_census.py now uses the same rule and j2_census.json agrees), the 32 surviving positions ranging over $r$ $0.86$–$0.93\,R_J$, latitude $64$–$73°$, longitude $265$–$301°$ (raw recaptures before the shell-floor cut: 36/40, reaching down to $r = 0.75$ — the two acceptance sets are stated separately on purpose). The two low-latitude nulls are markedly less persistent (15/40 and 19/40): under this chosen stress ensemble the polar root is the persistent one. The fractions move monotonically under paired amplitude scaling — the same 40 standardised coefficient draws scaled $\times\tfrac12/\times1/\times2$, so amplitude sensitivity is isolated from finite-ensemble variation: polar $35/40 \to 32/40 \to 29/40$, low-latitude $17\to15\to14$ and $28\to19\to15$. Widening the recapture radius from $0.10$ to $0.25$ moves 29/40 to 32/40; dropping the shell floor to $0.80$ raises it to 35/40 (members’ nulls sink below the floor rather than vanish). What would settle it is more Juno: the extended-mission orbits that motivated JRM33 keep tightening precisely those degrees. The prediction is on the record — a null at $r \approx 0.873\,R_J$, latitude $+69°$, longitude $282°$, with a dome footprinted on the patch boundary; survival fraction $32/40$ under the model-difference stress ensemble, sample position range (over those 32 survivors) $r\,0.86$–$0.93$, lat $64$–$73°$, lon $265$–$301°$ — and it is falsifiable by the next model release.

Two further limitations. The potential-field treatment of the envelope neglects any currents in the weakly conducting molecular hydrogen (standard practice, but an approximation — the same caveat as every solar potential extrapolation in this series); and the magnetodisc and magnetopause currents, which dominate Jupiter’s external magnetosphere, are irrelevant here — in the shell the $l \le 18$ internal field is of order $10^6$ nT (on a $1°$ grid: $1.8\times10^4$–$6.3\times10^6$ nT at $0.85\,R_J$, median $1.0\times10^6$; $1.8\times10^5$–$1.9\times10^6$ nT at $1\,R_J$; artifacts/j2_census.json) against their tens of nT, and it falls below that only in the immediate neighbourhood of the three nulls.

Where this leaves the program

Jupiter is the fourth world on which the growth-vector law has been checked ($Q\colon 5 \to 6$ on the nulls’ tangent cones), the fourth on which the vacuum/force-free theorem’s “radial only” prediction held, and the first where the series’ separatrix machinery answered a question about a planet’s interior: the polar patch is dome-covered parasitic flux, and the visible unipolar cap is the dome doing its job. The pattern that keeps repeating — on the Sun, at Earth, now inside Jupiter — is that the skeleton is a reproducible but resolution-fragile feature of the chosen model continuation: nulls and separatrices of modelled fields live and die by the least-constrained part of the model. The certificates and robustness protocols this series has accumulated are, increasingly, the actual product.

Glossary

• JRM33 / JRM09 — Juno Reference Model through perijove 33 (degree-30 fit, Connerney et al. 2022) / perijove 9 (2018). Spherical-harmonic models of the internal field; coefficients via the open planetmagfields package.
• System III — Jupiter’s magnetospheric rotation frame; longitudes here are the right-handed (east-positive) System III longitudes $\phi$ in which the JRM33 coefficients are defined; the traditional System III west longitude is $360° - \phi$.
• $R_J$ — Jupiter radius (71,492 km at 1 bar). The 1-bar surface ($r = 1$) is the visible cloud deck. The mapping surface $r = 0.85\,R_J$ is where JRM33’s standard field maps are drawn, near the top of the dynamo region; the Lowes-radius estimate places the conducting metallic-hydrogen dynamo top nearer $0.81\,R_J$ — “dynamo surface” in this post’s figures means the $0.85$ mapping surface.
• Parasitic polarity — a patch of reversed flux embedded in a dominant polarity region; generically covered by a null’s fan dome (the separatrix surface closing onto the polarity-inversion line) pierced by the spine (the null’s other eigen-direction, one foot inside the parasite).
• Polarity-inversion line — the $B_r = 0$ curve on a surface; here, the patch boundary.
• Gauss coefficients $g_l^m, h_l^m$ — Schmidt semi-normalised spherical-harmonic coefficients of the internal potential; JRM33’s dipole is 4.177 G tilted 10.25°.
• Growth vector $Q$ — the series’ sub-Riemannian null read-out (here a local confirmation at root-finder locations, not an independent detector): $Q = 5$ in bulk field, $6$ at a generic null (Parts 1–4).

Reproduce

cd research/preferred-directions
../cosmic-web/.venv/bin/pip install planetmagfields   # run used 1.7.0 (+ scipy 1.18.0,
                                                       # numpy 2.5.0, astropy 8.0.0)
../cosmic-web/.venv/bin/python src/jupfield.py               # golden checks
../cosmic-web/.venv/bin/python scripts/run_j1_jupiter.py     # the hunt + domes
../cosmic-web/.venv/bin/python scripts/run_j1b_dome.py       # per-patch scoring
../cosmic-web/.venv/bin/python scripts/run_j1c_baldpatch.py   # bald-patch PIL test
../cosmic-web/.venv/bin/python scripts/run_j1d_raw_w4.py      # non-circular w4 reading
../cosmic-web/.venv/bin/python scripts/run_j2_census.py       # cell-prefiltered census + ensemble
../cosmic-web/.venv/bin/python scripts/run_j2b_sensitivity.py # anatomy, degree check, floor + stress scans
../cosmic-web/.venv/bin/python scripts/render_j1_figures.py  # the four figures

Report: docs/J1-jupiter.md. Sources: Connerney et al. 2022 (JRM33) · Moore et al. 2018 (hemispheric dichotomy) · Zhang et al. 2021 (magnetospheric topology and the aurora) · planetmagfields.