Vortex

A vortex at $V$ rotates every point around the centre by an angle that depends on its radius, $\theta(r)$; the inverse lookup rotates by $-\theta(r)$. Because a rotation moves every point on a circle to another point on the same circle, the map is exactly area-preserving and can never fold, however large the angle. midi-sink ships two profiles.

Exponential (Jaffer)

$$\theta(r) = A\,e^{-r/R}$$

Shear concentrates near the centre and fades smoothly: diffuse, breath-like agitation. The default for the mod wheel, the Airwave dimensions and wind routing — “wind over the water”.

Exponential profile: A is the angle at the centre, R the e-folding radius. +A just off the top-left cluster, −A just off the bottom-right, ramped in n passes — off-centre on purpose: a rotation of circles about their own centre is invisible.open full size ↗

Rankine (rigid core)

$$\theta(r) = \begin{cases} \omega & r < R \\[4pt] \omega\,\dfrac{R^2}{r^2} & r \ge R \end{cases}$$

The core rotates rigidly — zero interior shear, so whatever pattern sits inside spins as a solid disk and survives intact — while all the shear lands in the crease ring at $r = R$ (the kink is intentional) and the $1/r^2$ fall-off outside. Mechanical, tight: the gesture literally grabs a disk of water and turns it.

Rankine profile, ω and −ω, each just off its cluster: the piece of the rings inside R turns as one rigid body, intact; all the shear lives in the crease ring at R and the 1/r² tail outside.open full size ↗

Invariants

• Exact at any angle. A finite rotation never folds, so neither profile is ever sub-stepped. Two passes of $A/2$ equal one pass of $A$.
• Area-preserving by construction.
• Rankine’s interior is exactly zero-shear: a ring cluster inside $R$ comes out bitwise as it went in, rotated.

Ownership

• CC 1 (mod wheel), Airwave CC 20 (Raise) → global vortex strength, exponential profile, through the CC map; CC 21/22 drift its centre.
• Two-finger twist (tablets, Marble mode) and right-drag (desktop) → Rankine, with $R$ = half the finger separation and $\omega$ = the twist delta; the settings’ Vortex profile selects which profile the gesture uses.
• Not pen orientation. Every attempt to stir a vortex from stylus tilt or azimuth leaked a phantom vortex (absolute orientation is noisy and has no zero) and was removed — orientation only ever feeds the vibrato booster, as a decaying derivative.

Lineage

The exponential profile is the vortex of Mathematical Marbling (Lu, Jaffer, Jin, Zhao & Mao, 2012). Rankine’s combined vortex is from his Manual of Applied Mechanics (1858) — see the citations. The soft, viscous sibling of both is the Lamb–Oseen swirl.