The Sun natural attractlet · § 7.4

Granules and Magnetic Field

Convection cells that sweep a magnetic field into the lanes between them, and a field strong enough to shut the cells down.

Take the heated layer from Convection Cells and thread it with a vertical magnetic field. The rising cells carry the field sideways and pile it into the sinking lanes between them, so the field at the surface gathers into thin, strong sheets while the cell centers are left nearly free of it. The Sun’s surface does this: its magnetic field sits in the lanes between granules. Turn the field up and it pushes back. Field lines resist being bent, so strong field slows the cells, and strong enough field stops them. Where convection stops, less heat gets through, and on the Sun that is why pores and sunspots are dark.

A Slice Through a Magnetized Layer

1 deep, 5.66 wide, repeating side to side · hot below, cool above

heating, Rayleigh number—

heating steady cells need with this field (field-free: 658)—

heat carried, times conduction alone—

strongest field at the surface, times the average—

share of surface field in its strongest tenth (10% is even)—

time (thermal diffusion times)—

cells forming

A real simulation, on a coarse grid, and a slice rather than a surface. It solves the two-dimensional equations of magnetoconvection on a 128 by 24 grid. Idealizations: free-slip walls, a Prandtl number of 1, magnetic diffusivity 0.3 times the thermal diffusivity, and none of the Sun’s sharp thinning with height. The Sun’s field gathers into far thinner, stronger sheets than a grid this coarse can hold. The field here is a sheet seen edge-on, not a spot. White lines are magnetic field lines; color is temperature, bright hot and dark cool. Time is the solver’s clock.

Why It Reads as an Attractlet

an authored reading, not a measurement

∂T/∂t + u·∇T = ∇²T
∂ω/∂t + u·∇ω = σ∇²ω + σR ∂T/∂x + σζQ (B·∇)∇²A
∂A/∂t + u·∇A = ζ∇²A
R heating · Q field strength · σ = 1 · ζ = 0.3 · A field lines

Same test as the Weather page. The cells carry heat that comes in through the bottom and release none of their own, as in Convection Cells. The field is carried too. The total field through the box is fixed from outside, and the cells only rearrange it; in two dimensions no flow can sustain field of its own against diffusion (Zeldovich’s antidynamo theorem). Heat and field are both handed in, so both the granules and the sheets of field read as a natural attractlet (§ 7.4). Press Cut the heat: the cells stop, and the field they gathered spreads back out.

Making field is the job of the dynamo, which is the next card and a harder reading.

Here the reading and the program agree: the layer is a natural attractlet, and the program that runs it is an attractlet model under § 7.4.

The Shape of the Basin

measured on this model, heating 20,000

Field gathers between the cells. With the field’s force switched off (strength 0), the surface field piles into thin sheets centered on the sinking lanes: the strongest tenth of the surface carries about 70% of it, and the peak is about 20 times the average. At strength 400 the cells are weaker, carrying 3.9 times the heat of conduction instead of 6.4, and the field collects in a narrow column between two cells where the flow nearly stops, a small cousin of a pore. Its tenth of the surface still carries about 70% of the field, with a peak about 9 times the average.

Strong field takes the basin. Cells ran at every strength tried up to 5,000, flickering in oscillating plumes from about 2,000 up. At 6,000 they had not started from a small disturbance after 2.5 thermal diffusion times, and a stir set off only slow growth. At 8,000 the still, magnetized layer held for 2 thermal diffusion times even after a stir. Two basins, then, divided by field strength: running cells with field gathered between them, and a still layer the field holds.

Inside a basin, the layer comes back. Press Stir it at strength 400: the cells return within half a thermal diffusion time, carrying the same heat. Press Cut the heat: the flow stops within about 0.3 thermal diffusion times, and the gathered field spreads back out, its peak falling from about 9 times the average to under 2 within about 1.2.

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