explorer — the collapse, in three dimensions

§2.1 · §3.1 · (3.2) · (4.3)–(4.7) · (4.25) · (4.29) · Theorem 4.6(v) of the paper · the section technique is bismuth's

Not boundary conditions — a choice of profile

The natural first guess for "a 3D solution explorer" is a fluid solver with boundary conditions you can pick. That is not what this problem is. The paper's solution lives in the whole space with a body force, and nothing is imposed at any boundary: the flow is designed, term by term, and the force is whatever residual the design leaves. What is free in that design is the profile — the functions E(X, η) and U(X, η) that give the swirl and the axial flow in similarity coordinates — and the paper spends its appendices proving a profile exists with all the properties it needs, without ever writing one down. So the generative object here is the profile: a seed picks one member of a family that has every structural property the paper demands, and everything else — the meridional flow, the pressure, the exterior, the collapse — follows from it by the paper's own identities.

What is exact and what is modelled

The field on screen is u_θ = q^−AE, u_z = q^−AU, r·u_r = V₀, p = q^−2AΠ in the similarity coordinates τ = q(1 − η²), z = q^Dη, X = r²/2q of (3.2), with A = ½ + h and D = ½ − h. Exact, and pinned by the selftest:

• Incompressibility. V₀ is computed from U by the paper's identity (4.7), so ∇·u = 0 to finite-difference precision everywhere, at every time.
• Zero axial moment. U has a return shell chosen so ∫₀^∞U dX = 0 exactly — the paper's M(∞, η) = 0 in (4.28) — which is what makes the radial flux vanish outside the core instead of pulling fluid in from infinity.
• Regularity at the axis. E = √(2X)·F with F smooth, so the swirl and radial velocity vanish linearly at the axis and the Cartesian field (4.5) is smooth.
• The pressure. Π = −∫_X^∞E²/2x dx, the leading centrifugal balance normalised to vanish at infinity (4.25): the low pressure you see is exactly what holds the swirl in.
• The exterior. Far from the axis E → c_∞X^−AH(2d/X), the heat-law swirl of the exterior page, and the flow becomes purely azimuthal.
• The scalings. Radius ∼ τ^½, height ∼ τ^½−h, speed ∼ τ^−½−h, and u_z = 0 on the layer η = −j₀/4 below the middle plane — all the numbers of the core page, live.

Modelled: the shape of E and U between the axis and the far field (a Gaussian axial core with a Gaussian return shell; a swirl that rises linearly, bulges by a factor 1 + β/(1 + η²) at intermediate radius and settles onto the heat law), and the radii of the pulse annulus. The paper's actual profiles are built by matching five radial moments across joins and adjusting the shear to sit inside the stress cone; ours do not satisfy the momentum balance in the annulus and are not meant to — that residual is exactly what the paper's pulses exist to cancel. The rings you can switch on mark where they would live; they are decoration.

Why the streamlines hardly change

Run the collapse and the streamlines shrink with the core but keep their shape; in the follow frame they stand almost still while the readouts run away. That is not a rendering shortcut — it is what self-similar means, and it is the content of the theorem. In similarity coordinates the velocity is u_θ = q^−AE(X,η), u_z = q^−AU(X,η), r·u_r = V₀(X,η): the profiles E, U, V₀ do not depend on time at all, so the direction of the flow at a given similarity point is fixed and its magnitude is what grows. Streamlines are direction fields, so they are frozen in similarity units and shrink like √τ in the fixed frame; the flux through them — speed times area — is what changes, as τ^−½−h times τ. Pathlines are another matter: a particle rides a field that is strengthening under it, which is why the particle cloud does evolve.

The one thing that does drift, slowly, is the mix of components. Swirl and axial speed scale like q^−A = q^−½−h, the radial inflow like q^−½, so the swirl-to-inflow ratio grows like τ^−h — the panel shows it. With h under 1/100 that is a few percent across three decades of τ: streamlines wind very slightly more turns per unit of inward progress at late times. In the paper itself this is the only time dependence of the leading field too; everything else — the corrections in powers q^2h, the pulses on their dyadic bands — is smaller by positive powers of q.

How it is drawn

The slice is the trick from bismuth's magnetism view, turned to face you: a plane perpendicular to the line of sight at a depth you choose, with everything between you and it clipped away in the fragment shaders, so you look into the object rather than at its front. On the plane, pressure is colour on a log scale, and direction is line-integral convolution: white noise smeared along the in-plane velocity, eight steps each way, faded where the flow runs through the plane rather than along it. Looking down the axis it shows the swirl as spirals; from the side, the inflow and the axial jets. In the fixed frame the colour scale is absolute — the core value at τ = 1 — so the core saturates to white as it deepens; following the core it is relative to the current core value. Behind it, streamlines of the frozen field at the current τ are integrated on the CPU in similarity coordinates and rescaled with q, drawn additively; particles are seeded by rejection sampling with acceptance proportional to the local speed (so the slow exterior stays nearly empty and the core is dense), then advected on the GPU by transform feedback with the same field, in physical time — while τ runs they trace genuine pathlines of the time-dependent flow, recruited at the current scale as the core shrinks; and the glow is a ray march through the pressure field, emission rising steeply with suction. The field is written once in JavaScript and once in GLSL from the same formulas, and both read the same two tables — H(Z) for the heat factor and Π(X, η) for the pressure — uploaded as float textures.

Two honest limits. In the fixed frame the object shrinks like √τ, so by τ ≈ 10⁻³ it is a few pixels; that is the point, and "follow" is the antidote. And with h < 1/100 the picture in similarity units barely changes with τ — the ratios that do change go like τ^−h — so following the core is very nearly watching a still image while the speeds and the pressure run away underneath it. That, too, is the content of the theorem.