Scales and exponents

One small exponent $h$ fixes every rate in the construction. This page lists the rates and lets you move $h$.

Write $\tau = 1-t$. The core is a column whose radius shrinks like $\tau^{1/2}$, the viscous length, and whose height shrinks more slowly, like $\tau^{1/2-h}$. The anisotropy is the point: it is what lets the speed grow faster than $\tau^{-1/2}$ while the energy still vanishes.

Similarity variables

The paper fixes $0 < h < 1/100$ and writes $A = \tfrac12 + h$ and $D = \tfrac12 - h$. In cylindrical coordinates $(r,\theta,z)$ it introduces a concentration scale $q>0$ and an axial variable $\eta\in(-1,1)$ through

$$\tau = q\,(1-\eta^2),\qquad z = q^{D}\eta,\qquad X = \frac{r^2}{2q}.$$

On a fixed compact set of $(X,\eta)$ away from $\eta=\pm1$ one has $q\asymp\tau$, so $q$ is the time remaining, reparametrized so that the axial coordinate has its own rate. The leading axisymmetric field is built from three profiles $E$, $U$, $\Pi$ of $(X,\eta)$:

$$u_\theta^{(0)} = q^{-A}E,\qquad u_z^{(0)} = q^{-A}U,\qquad r\,u_r^{(0)} = V_0,\qquad p^{(0)} = q^{-2A}\Pi .$$

In English: swirl and axial speed grow like $q^{-1/2-h}$, the radial inflow is weaker by a factor $q^{h}$, and the pressure is the square of the speed. Regularity on the axis is a condition on the profiles, $E/\sqrt{2X}$ smooth at $X=0$, and incompressibility fixes $V_0$ from $U$.

The rates

Everything below follows from the two length exponents and the one speed exponent. The demo plots them.

Quantity	Scale	What it does as $\tau\downarrow 0$
core radius $\ell_r$	$\tau^{1/2}$	the viscous length; shrinks
core height $\ell_z$	$\tau^{1/2-h}$	shrinks, but slower: $\ell_r/\ell_z\asymp\tau^{h}\to 0$
swirl, axial speed	$\tau^{-1/2-h}$	unbounded
radial speed	$O(\tau^{-1/2})$	unbounded, but smaller by $\tau^{h}$
core volume	$\tau^{3/2-h}$	vanishes
core energy	$\tau^{1/2-3h}$	vanishes, since $h<1/6$
core dissipation rate	$\tau^{-1/2-3h}$	diverges but is integrable in time
angular Reynolds number	$\tau^{-h}$	diverges, slowly
radial Reynolds number	$O(1)$	viscosity stays in the leading balance
axial / radial diffusion	$\tau^{2h}$	the expansion parameter of the background corrections

What to look for: the speed line rises and the energy line falls for every $h$ below $1/6$, and the two length lines separate. Push $h$ toward $1/6$ and the energy line flattens; past it the energy would grow and the theorem's bounded-energy claim would fail. The paper keeps $h<1/100$, far from that edge, because the correction cycle needs room, not because the energy does.

Why the anisotropy

With $h=0$ the core would be an isotropic self-similar vortex with speed $\tau^{-1/2}$. Such a flow has $\int|u|^3$ bounded, and by Escauriaza, Seregin and Šverák that cannot blow up without a force, and by the scaling argument of Nečas, Růžička and Šverák there is no nontrivial self-similar profile at all. A positive $h$ is a way past both. The speed exceeds the critical rate $\tau^{-1/2}$ by $\tau^{-h}$, the column is slender enough for the energy $\tau^{1/2-3h}$ still to vanish, and the axial direction supplies a second small parameter, $\ell_r^2/\ell_z^2 = \tau^{2h}$, in which the background can be corrected order by order.

The angular Reynolds number $\tau^{-h}$ says the fluid makes ever more turns in one radial diffusion time. That is what amplifies the pulses at the edge of the core. The radial Reynolds number $O(1)$ says viscosity never drops out of the radial balance. That is what makes this a Navier–Stokes construction rather than an Euler one dressed up.

Viscosity is a change of units

The construction is done at $\nu = 1$ and transferred to any $\nu>0$ by

$$u_\nu(x,t) = \sqrt{\nu}\,u(x/\sqrt{\nu},\,t),\qquad p_\nu = \nu\,p(x/\sqrt\nu,t),\qquad f_\nu = \sqrt{\nu}\,f(x/\sqrt\nu,t).$$

Time is untouched, so the singular time stays at one. Lengths scale by $\sqrt\nu$ and speeds by $\sqrt\nu$. The energy scales by $\nu^{5/2}$. This matters for the molecular limit: the core radius in metres is $\sqrt{\nu\tau}$ times a fixed number, and the core speed is $\sqrt{\nu/\tau}$ times a fixed number, so their product is $\nu$ times a fixed number for every fluid.

Next: the core.