The core

A vortex that spirals inward, spins up by conservation of angular momentum, and expels fluid along its axis.

The leading flow is axisymmetric and lives in three regions: a concentrating inner core, an annulus where the core is joined to the outside, and a heat exterior. The core is where the speed diverges. This page is about why it can.

Spin-up

A fluid parcel that feels no torque conserves its angular momentum per unit mass, $r u_\theta$. Carry it inward and its swirl grows like $1/r$. This is the bathtub vortex, and it is the only amplifying mechanism the construction needs in the core. Inflow carries angular momentum toward the axis; viscosity diffuses it back out; the balance of the two sets the rate at which the characteristic speed grows.

Inflow cannot continue unless the fluid goes somewhere. Incompressibility forbids piling up on the axis, so the core has axial outflow: fluid arriving at small radius leaves upward on one side of a dividing layer near $z=0$ and downward on the other. The pressure gradient points toward the axis, supplying the centripetal force for the swirl, and its axial component points toward the mid-plane.

Symmetry, deliberately broken

The obvious profile would be reflection-symmetric in $z$. The paper does not use it. On the mid-plane $z=0$ an exactly symmetric axial flow vanishes, and with it the radial shear of axial velocity that the pulses in the annulus need for their amplification. Away from the mid-plane, axial transport of angular momentum supplies a steep radial fall in angular velocity, and the rotational mechanism amplifies pulses on its own. The profile is therefore given a small upward bias with nonzero axial velocity at $z=0$. This separates the layer where rotational amplification is weak from the layer where axial shear vanishes, so that at every height at least one mechanism is active.

The profile equations

In the similarity variables of the scales page, the leading velocity is

$$u^{(0)} = \frac{V_0}{r}\,e_r + q^{-A}E\,e_\theta + q^{-A}U\,e_z,\qquad p^{(0)} = q^{-2A}\Pi,\qquad \Pi(X,\eta) = -\int_X^\infty \frac{E(x,\eta)^2}{2x}\,dx .$$

The pressure formula is the radial balance $\partial_r p = u_\theta^2/r$, integrated in from infinity. The radial flux $V_0$ is fixed by incompressibility from $U$. What remains is to choose $E$ and $U$ so that the tangential momentum equations, azimuthal and axial, hold to leading order in the core. The paper solves these near the axis as an analytic problem in $(X,\eta)$, with $E/\sqrt{2X}$ and $U$ smooth at $X=0$, which is what makes the Cartesian field smooth on the axis for every $t<1$. Regularity on the axis is not automatic in cylindrical coordinates and the construction pays for it throughout: every correction is built from vector potentials whose curl is smooth there.

A single displayed fact carries the theorem. For a fixed $X_{\rm in}$ inside the core,

$$u_\theta\!\left(\sqrt{2X_{\rm in}\tau},\,0,\,0,\ 1-\tau\right) = \tau^{-A}\big(e_0 + O(\tau^{2h})\big)\longrightarrow +\infty,\qquad e_0>0 .$$

In English: on the circle of radius $\sqrt{2X_{\rm in}\tau}$ in the mid-plane, the swirl grows like $\tau^{-1/2-h}$ with a definite positive coefficient. Everything added later, pulses and corrections, vanishes in this fixed inner region or is smaller by powers of $\tau$. The localization cutoffs equal one there. So the final velocity inherits this growth, and this is the line the proof of Theorem 1.1 points to when it says the solution is unbounded.

The heat exterior

Beyond the annulus the flow is pure swirl, independent of height: $u = K(r,\tau)\,e_\theta$ with

$$-\partial_\tau K = \partial_r^2 K + r^{-1}\partial_r K - r^{-2}K,\qquad K(r,\tau) = r^{-1-2h}H_{\rm ext}(\tau/r^2).$$

This is the azimuthal component of the Stokes equations, a radial heat equation for the swirl. It is solved exactly, so its residual is zero and no force is needed there. Its profile decays like $r^{-1-2h}$ at large radius, matching the core's $E\sim c_\infty X^{-1/2-h}$, and at every fixed positive radius the field and all its derivatives have smooth limits as $\tau\downarrow 0$. That last property is what allows the whole flow to be cut off smoothly in space without disturbing the concentrating core, and it is why the singular point is isolated.

The annulus is the problem

Joining an inner profile that satisfies its own balance to an exterior that satisfies a different one leaves a mismatch. Write the tangential residuals of the leading field as $R_\theta^{(0)}$ and $R_z^{(0)}$, retaining radial viscosity and omitting axial viscosity, which is smaller by $\tau^{2h}$. The paper writes them as divergences of a stress:

$$R_\theta^{(0)} = -(\partial_r + 2/r)\,T_{r\theta},\qquad R_z^{(0)} = -(\partial_r + 1/r)\,T_{rz},$$

with $T$ obtained by integrating the residuals in from the axis. Five radial moment conditions, matched at the join, make $T$ vanish inside the core and outside the exterior radius, so the stress is supported in an annulus $X_a < X < X_b$ and nowhere else. The background residual there scales like $q^{-3/2-h}$ and does not extend through $\tau=0$. It has to be supplied by something that is not an external force. The next page is about the something.

Next: the annulus and the pulses.