The Sun reading open

The Sunspot Cycle

Two parts of the Sun’s magnetic field, each converted into the other by the Sun’s flows, on a cycle of about eleven years.

The Sun’s magnetic field has two parts. Its rotation changes with depth and latitude, and that shear stretches north-south field lines into east-west bands. Twisting convection lifts and turns loops of those bands, and makes new north-south field. Each part of the field is converted into the other, by a different flow in each direction. Parker showed in 1955 that this loop runs as a wave: bands of field form, travel across latitude, and reverse. The field’s two parts regenerate each other, but the flows that do the converting are supplied to the field, not made by it. Sunspots appear where the east-west field is strong, so on the Sun the wave shows as the butterfly diagram: spots start at mid latitudes and drift toward the equator over about eleven years, with opposite magnetic signs in the two hemispheres (Hale’s law), and the signs swap each cycle.

The Field, Pole to Pole, Over Time

east-west field by latitude · red one sign, blue the other · newest at the right

dynamo number D, twisting times shear (critical |D| for this model ≈ 275)—

sunspot cycle, years between reversals—

strongest east-west field (1 = strong enough to weaken the twisting)—

north and south hemispheres—

bands travel—

time (years)—

starting

A real calculation of a classic model, far simpler than the Sun. It solves a mean-field dynamo of Parker’s kind in one dimension, latitude, for a thin shell, on 121 points. The twisting by convection and the shear are prescribed, not simulated; strong field weakens the twisting, and that is what stops the growth. In the real Sun the field also acts back on the rotation and convection through its Lorentz force; here it does not. The field has no depth here, and no flow carries it. The colors show the east-west field, not sunspots. Time is scaled so the default setting gives an 11-year cycle, so the model does not predict the length. Its pattern is steeper than the Sun’s: a band of field here crosses from 35° to 10° latitude in about 2 years, while the Sun’s belt of spots starts above about 20° and takes most of a cycle to reach the equator.

The Loop, in Two Equations

Parker’s dynamo wave, thin-shell form

∂A/∂t = cosθ · B/(1 + B²) + ∂²A/∂θ² − A
∂B/∂t = D ∂(sinθ A)/∂θ + ∂²B/∂θ² − B
A north-south field · B east-west field · θ angle from the north pole · D dynamo number

The loop is the first term of each equation. In the first, east-west field B makes north-south field A through the twisting of convection, cos θ, which turns one way in the north and the other way in the south. In the second, north-south field makes east-west field through the shear, D. The other terms are losses. Neither part can grow alone: set D to zero and both fade. The default is D = −400, so |D| = 400; the slider sets |D| and the rest of the page quotes |D| unless a sign is given.

The factor 1/(1 + B²) is the only thing the field does back to what drives it: strong field weakens the twisting. Here the field never changes the shear or drives the convection. In the real Sun it does act back on both, which is one reason this model is not a model of the Sun’s whole dynamo.

Why the Reading Is Open

an authored reading, not a measurement

Three things, classified separately. The program that runs the panel is an attractlet model under § 7.4, as on the Lorenz page. The field as this model has it is a second thing. The Sun’s dynamo is a third, and this page does not classify it.

Name the boundary first. Take X to be the magnetic field alone, its two parts A and B, with the twisting and the shear supplied from outside X. That is exactly what the two equations contain.

Regeneration is not continuation. Inside that boundary a loop closes: B is converted into A, and A into B. But each conversion is done by a flow the field does not build or sustain. The field closes a loop of state conversion, not a loop of continuation. Recursion Lock (§ 7.2, Condition 1) asks for continuation through closed-loop feedback without external orchestration, and here the machinery of both conversions is supplied from outside X. Read that way, the field as modeled is an attractlet: its apparent persistence is “externally sustained rather than internally produced” (§ 7.4). The field does draw more energy from the flows when it is stronger, but drawing more power from a source is not the same as maintaining oneself.

The principle this page uses: closure of state conversion is not closure of continuation. A loop of states counts toward recursion only if the machinery that performs each conversion lies inside the candidate’s own continuation. If that machinery is supplied from outside, the loop can grow, hold a cycle and recover from a wipe, and still be an attractlet. This is an authored reading principle for this page, not kernel canon.

Why the tag stays open. The tag reads the Sun’s cycle, not this model, and for the Sun the boundary is not settled. Widen X to the coupled system of rotation, convection and field, and the field’s Lorentz force acts back on the flows that convert it. Whether that larger X closes its own continuation is a separate question, and a model that prescribes the flows cannot answer it.

The Shape of the Basin

measured on this model

A threshold. Below |D| of about 275 the field fades: at |D| = 250 a seed dies away, and at D = 0 a full-strength field falls to a tenth in about 27 years. Above it, a faint seed grows. The number 275 belongs to this model’s scaling, geometry, boundaries, quenching law and grid; it is not a general constant of Parker’s dynamo.

One cycle, whatever the seed. At the default D = −400, four different random seeds, and a seed placed in the north alone, all grew into the same cycle: a reversal every 11 years, the strongest field 1.0, bands strongest on average near 24° latitude and moving toward the equator, and opposite signs in the two hemispheres. In this symmetric model even a one-hemisphere seed converges to that Hale-like pattern of opposite signs; it follows from the model’s symmetry and coupling, and does not reproduce the Sun’s mechanism for Hale’s law. Press Wipe to a faint seed and the field is back to 90% of its strength in about 335 years.

Stronger loops, faster cycles. At |D| = 300 the field peaks near 0.4 and reverses every 11.2 years; at 600, 1.6 and every 10.0 years; at 750, the top of the slider, 1.8 and every 9.0 years, still with opposite signs north and south.

The Sign Problem

where this model and the Sun disagree

Parker’s waves travel toward the equator only when the twisting and the shear have opposite signs, the Parker-Yoshimura sign rule, so this model uses a dynamo number below zero. Helioseismology finds that, at the base of the convection zone and at the low latitudes where spots appear, the shear has the sign that by the same rule sends waves toward the poles.

Press Flip the shear to give the model that sign. This model does not even make poleward waves; its cycle ends. The last reversal comes about 26 years later, and within about 50 years the field has settled into a steady field about five times stronger, strongest near 50° latitude. With the shear flipped, field grows once D is between +20 and +40, but it never cycles.

Many current models add a slow flow, toward the poles near the surface and toward the equator at depth, to carry the field instead. There is still no consensus on how the solar dynamo works.

Parker, E. N. (1955). Hydromagnetic dynamo models. The Astrophysical Journal, 122, 293.
Yoshimura, H. (1975). Solar-cycle dynamo wave propagation. The Astrophysical Journal, 201, 740.
Hathaway, D. H. (2015). The solar cycle. Living Reviews in Solar Physics, 12, 4.
Charbonneau, P. (2020). Dynamo models of the solar cycle. Living Reviews in Solar Physics, 17, 4.

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