Lamb–Oseen swirl

midi-sink’s founding rule bans Navier–Stokes solvers. Here Navier–Stokes enters anyway — as an exact solution. The Lamb–Oseen vortex is the closed-form decay of a 2-D line vortex in a viscous fluid, with tangential velocity $v_\theta(r) = \frac{\Gamma}{2\pi r}\left(1 - e^{-r^2/r_c^2}\right)$. Integrated over the pass interval it becomes a rotation angle:

$$\theta(r) = \frac{\Gamma\,\Delta t}{2\pi r^2}\left(1 - e^{-r^2/r_c^2}\right), \qquad \text{rotate } P \text{ about the centre by } -\theta(r).$$

One voice struck softly at the centre and covered with a clear drop — the core is water, nothing marks it — and four ring pools stirred for t frames: the 1/r² far field winds them into spirals. Γ from the pressure amount, r_c from the strike velocity (far from the core the field does not depend on it).open full size ↗

Invariants

• A pure rotation — exactly area-preserving and, like the Rankine, an exact map at any angle: no sub-stepping, ever.
• $C^\infty$. Where the Rankine has its crease-ring kink at $R$, Lamb–Oseen is smooth everywhere: solid-body rotation in the core blending viscously into the same $1/r^2$ far field. The soft, organic twist beside the Rankine’s mechanical one.
• Numerical guard. The expression is $0/0$-shaped at small $r$ and cancels in floating point; the shader evaluates it through expm1, and below $r \ll r_c$ uses the limit $\theta(0) = \Gamma \Delta t / (2\pi r_c^2)$ directly.
• Half-float freeze, accepted. In RGBA16F, sub-quantum per-pass displacements near the exact centre round away — protective, never inflating. Field measurements at $r \to 0$ read low, not high; this is not a bug to chase.

Ownership

• Polyphonic key pressure (0xA0) on the voice’s note → swirl, unconditionally, in either pressure mode. No reference hardware sends it; the tablets’ Play surface does — pull the finger back (Y down) and the note stirs.
• The Marble-mode pressure gesture (v0.6): a long press (Shift + right drag with a mouse) lays a drop; pull back and sumi_add_vortex with the Lamb–Oseen profile stirs around it, the drop as its core — the same pass the voice mapper emits, with the host supplying Γ·Δt and r_c.
• Channel pressure (0xD0) with press_mode = 1 → swirl instead of the ink feed, so pressure-only hardware (ROLI, Osmose) can reach it. One consumer owns 0xD0.
• The drop is the vortex core: $r_c$ = the voice’s current nominal boundary $R$, so its own rings rotate near-rigidly and stay coherent while the far field stirs the neighbours. $\Gamma$ per pass ∝ smoothed amount × Δt (a delta rate, like every feed); the rotation sign defaults to the voice’s band parity, so adjacent notes counter-rotate with zero configuration. Swirl passes fan out to every echo and count against the frame budget like any feed.

Lineage

This is Aubrey Jaffer’s own extension of his closed-form family: The Lamb–Oseen Vortex and Paint Marbling (arXiv:1810.04646, 2018) derives the displacement pattern of a decaying Lamb–Oseen vortex for marbling. midi-sink arrived at the same operator independently, from the per-note pressure dimension looking for a stir; finding his paper afterwards was the project’s happiest discovery. The vortex itself is in Lamb’s Hydrodynamics — see the citations.