Burst

Method after A. Jaffer, “The Lamb–Oseen Vortex and Paint Marbling” (arXiv:1810.04646), extended here to the multipoles m ≥ 2.

Strike the water with an impulse that has no net push and no net spin — a quadrupole — and four lobes form: two eject along an axis, two draw in across it. In a viscous sheet the impulse’s vorticity diffuses as it acts, and the flow it drives dies on its own clock. The burst is the whole of that motion, integrated over time into one displacement, the way Jaffer integrated the decaying Lamb–Oseen vortex into a single closed-form turn.

The integral closes. The $m$-th viscous multipole has the stream function $\psi_m \propto \gamma_m(s)\,\sin(m\theta)/r^m$ with the cutoff $\gamma_m(s) = 1 - e^{-s}\sum_{k<m} s^k/k!$ and $s = r^2/4\nu\tau$ — the Lamb–Oseen vortex is $m = 0$, the viscous stroke’s Stokeslet $m = 1$. Integrating over the life of the flow,

$$\int_0^t \gamma_m\!\left(\tfrac{r^2}{4\nu\tau}\right) d\tau = \frac{r^2}{4\nu}\,\Phi_m(S),\qquad \Phi_m(S) = \int_S^\infty \frac{\gamma_m(s)}{s^2}\,ds,\qquad S = \frac{r^2}{4\nu t},$$

and by parts $\Phi_m = \gamma_m(S)/S + \Gamma(m-1, S)/(m-1)!$. For the dipole the remainder is the exponential integral $E_1$ — the logarithmic kernel that makes the stroke special. For every $m \ge 2$ it is an exponential polynomial, and $\Phi_m$ is elementary; for the quadrupole, $\Phi_2 = \chi(S) = (1 - e^{-S})/S$, the stroke’s own $\chi$. With the impulse spread over a Gaussian core of radius $a$ and aged to $\ell$ ($\ell^2 = a^2 + 4\nu t$),

$$\Psi = A_m \left(\tfrac{a}{r}\right)^{m-2} \sin\!\big(m(\theta-\theta_0)\big) \left[\Phi_m\!\left(\tfrac{r^2}{\ell^2}\right) - \Phi_m\!\left(\tfrac{r^2}{a^2}\right)\right], \qquad \mathbf d = \nabla^{\perp}\Psi .$$

Where the pinch comes from. Near the core the quadrupole’s displacement is pure hyperbolic strain, $d_x = \lambda x,\ d_y = -\lambda y$: the pinch is this burst’s $r \to 0$ limit, as the Lamb–Oseen swirl’s core is solid-body rotation. In the diffused zone $a \ll r \ll \ell$ the quadrupole decays as $\cos 2\theta / r$ (order $m$ as $\cos m\theta / r^{m-1}$); beyond $\ell$ every order falls as $1/r^{m+1}$, the potential multipole the vorticity has not yet reached. The burst is to the pinch what Lamb–Oseen is to solid-body rotation: the physical bridge between the core and the far field.

The strike has a lifetime. It fires sharp at $\ell = a$ and the release grows the age to its final value over the life; $\ell^2$ grows linearly in time, so most of the motion lands early and the tail is soft — the discharge blooms and dies. The amplitude $D$ is the displacement of the lobe at $r = a$ over the whole burst; the age shapes the strike without scaling it. $\theta_0$ aligns the ejection lobes — the pen’s azimuth, or the glide direction, where the medium’s binding tables route the strike.

Two clusters, then two strikes: A along θ₀, B a quarter turn on. D is the lobe displacement in units of the core, a the core, m the order (2 the quadrupole, 3 three lobes), ℓ/a how far the discharge diffuses, τ the release in seconds.open full size ↗

Invariants

• Sub-stepped. $\mathbf d$ is divergence-free as a field; the finite map preserves area to first order. The core splits every burst into passes whose peak displacement stays within a fraction $\beta_m$ of the pass’s current core — the wake’s $a/4$ criterion ($|\nabla \mathbf d| \le 0.25$) measured for each order — and one such pass reads a pre-image Jacobian of at least 0.81 everywhere with mean 1.00000 on the desktop harness. A burst and its negative leave the first-order residual, a fraction of a texel at the core.
• The stagnation origin. The centre does not move; the lobes at $r = a$ move by exactly $D$, outward on the axis, inward across it.
• The shader is the closed form. Along the axis the desktop harness reads the displacement at $a/2$, $a$, $2a$, $3a$, $4a$ to a fortieth of a texel of the double-precision reference.
• The age envelope. At a sixth of the release the lobe has moved the share $[\Phi_2(1/3.5) - \Phi_2(1)]/[\Phi_2(1/16) - \Phi_2(1)] = 0.70$ of $D$ (age 4); at the end, $D$, less the medium’s half-float quantum.
• The ingress rule (fresh water where a pass reaches beyond the canvas) applies as to the wake.

Ownership

• sumi_add_burst(x, y, a, D, θ₀, m) — the gesture: it starts the episode whose increments the engine emits over the release. Age (the final diffusion length over the core, 1.5–12) and Life (the release in seconds; 0 fires the whole burst at once). A negative $D$ is the first-order inverse. The order clamps to 2–8.
• Inside the Anod strike. Under the Anod medium a note’s strike is the spark composition, and the burst is its middle piece: a burst of the strike’s charge — core $0.3\,r$ — with its order from Order (burst_order, 2–8, default 2) and its lobes along the note’s axis. The per-pitch-class order table stays in the ABI, serialized and unused: a composition that wants it has its table.

A note on where this formula came from

I wasn’t looking for a theorem — I was looking for a spark. The strikes in the electric medium felt too round, too liquid, nothing like the discharge I see when the music peaks. So I did what this whole project does: I followed Professor Jaffer’s method. His Lamb–Oseen paper integrates a viscous flow over time to get a closed-form displacement; I tried the same integration on the higher multipoles, expecting special functions — and for m ≥ 2 the integral simply closed. The logarithmic kernel that haunts the dipole vanishes, and everything becomes elementary. I searched the literature as well as I could and did not find it stated in this form, but I claim nothing: serendipity did the work, and Jaffer drew the map. I only needed the discharge to bloom sharp and die soft, the way it sounds in my head. Here is its closed form.

— the author

Lineage

The viscous multipoles — the Stokes-approximation solutions of the vorticity diffusion equation for a localized impulse, the Hermite modes of a diffusing vortex — are classical fluid mechanics (Voropayev & Afanasyev; Chan & Chwang’s unsteady singularities; Gallay & Wayne). Aubrey Jaffer’s integration of the decaying Lamb–Oseen vortex into one closed-form displacement (2018) is the method, applied here to the orders above the dipole. The wake’s viscous stroke is the $m = 1$ case, with its $E_1$; the pinch is the quadrupole’s core; the algebraic lobe the early roadmap sketched is superseded by the burst and never shipped. The literature check behind the wording is in the citations.