The annulus and the pulses

Oscillations with zero mean carry momentum with nonzero mean. The construction aims that momentum at the residual.

The background residual in the annulus is a stress divergence, singular as $\tau\downarrow 0$. An external force that supplied it would be singular too. So the paper adds a sequence of spatially oscillatory pulses, seeded by an exponentially small force and grown by the background shear, whose averaged momentum flux is the stress the background needs.

Zero-mean waves, nonzero-mean flux

Consider fluid at the edge of the core moving outward with a surplus of swirl, and fluid moving inward with a deficit. Both carry angular momentum outward. Reverse both signs and the product $w_r w_\theta$ is unchanged. A pattern of such motions around a complete ring has zero angular average in every component and a definite angular average in its radial flux of azimuthal momentum. The same holds for axial momentum. That is the quantity the annulus is missing, and the leading covariance identity of the construction reads

$$\begin{pmatrix}\langle \tilde w_r \tilde w_\theta\rangle\\ \langle \tilde w_r \tilde w_z\rangle\end{pmatrix} = T + \text{higher order},$$

where $T=(T_{r\theta},T_{rz})$ is the annular stress of the core page and the bracket is the angular average. The radial divergence of the left side cancels the leading residual of the background. Note the sizes: the pulse amplitude is $A_{\rm wave}\asymp q^{-1/2-h/2}$, smaller than the background speed $q^{-1/2-h}$ by $q^{h/2}$, but its square divided by the wavelength $q^{1/2+h/2}$ is $q^{-3/2-h}$, exactly the scale of the background residual. The pulses are a lower-order velocity and a leading-order stress.

Two families and a cone

The stress has two components and one wave gives one flux direction. So each patch of the annulus carries two families of pulses with different phase gradients, with covariance directions $v_1, v_2\in\mathbb{R}^2$ that depend on the local background. Squared amplitudes are nonnegative, so the stresses two families can produce are the cone $\{c_1 v_1 + c_2 v_2: c_1,c_2\ge 0\}$. The profiles $E$, $U$ of the core are built so that $T$ lies strictly inside this cone wherever it is nonzero. That is the admissible stress cone condition, and it is a constraint on the core imposed by the annulus.

The admissible stress cone. The coefficients stay positive under small perturbations of $v_1$, $v_2$, which is what lets later corrections of either sign be realized by linearizing at the fixed amplitudes.

Growth, then decay

A pulse is a wave $a\cos(\xi\cdot x+\varphi)$ with $a\cdot\xi = 0$, oscillating across its own wavevector. In the moving frame of the background it obeys a two-dimensional amplitude equation along the pulse coordinate $v$,

$$z' = \big(\mathrm{diag}(\lambda(v),-\lambda(v)) + E\big)z - d(v)\,z,$$ $$\lambda(v) = \frac{\lambda_0}{\sqrt{1+s(v)^2}},\qquad d_{\rm ref}(v) = \varepsilon k^2 B_s^2\,(1+s(v)^2),$$

with $|E|$ small. The growing coordinate extracts energy from the shear at rate $\lambda$; the damping $d$ is viscosity acting on the squared wavenumber. The shear tilts the wavevector as the pulse rides along, which is the linear growth of $s(v)$: tilting weakens the extraction and lengthens the wavevector, so the damping overtakes. The net rate $\lambda - d_{\rm ref}$ is positive at first and negative later, and vanishes at the midpoint by construction. The envelope

$$P(v) = \exp\!\int_{L_s/2}^{v}\big(\lambda(w)-d_{\rm ref}(w)\big)\,dw,\qquad e^{-C(v-L_s/2)^2/L_s}\le P(v)\le e^{-c(v-L_s/2)^2/L_s},$$

is Gaussian in $v$: exponentially small at both ends of the pulse interval. That is what makes each pulse cut off cleanly, and it is the sense in which each pulse is seeded by an exponentially small force. The energy budget of the pulse is exact:

$$\frac12\frac{d}{dv}|t|^2 = -\,g\cdot\big(t_r(t_\theta,t_z)\big) - \varepsilon k^2|n_\Phi|^2\,|t|^2 .$$

In English: the pulse gains energy from the background shear $g$ through the very momentum flux $t_r(t_\theta,t_z)$ that it is there to deliver, and loses it to viscosity. The mechanism that supplies the stress is the mechanism that feeds the pulse. This is Orr's mechanism from shear-flow stability, and the exact waves of Craik and Criminale are the inviscid ancestors.

What to look for: the growth rate falls monotonically and the damping rises monotonically along the pulse, they cross once, at the midpoint, and the envelope peaks there. Lengthening the pulse deepens the Gaussian tails at both ends, which is the seed becoming smaller. If any parameter produced two crossings or a peak away from the midpoint, the envelope bound above would be false; it cannot, because $\lambda_0$ and $B_s$ are tied by the midpoint condition.

Keeping the pulses apart

Pulses on successively finer scales must not interact except where the construction wants them to. The paper gives each pulse an auxiliary variable $Y$ on a torus $\mathbb{T}^2$, computes covariances on that torus, and only afterwards evaluates $Y$ at a fixed phase map $Y(r,t)$. Pulses whose supports overlap in physical space are given disjoint supports in $Y$, so their products vanish identically. Harmonics of the same pulse keep their interactions, and those are what the correction cycle handles. Exact incompressibility comes from taking curls of supported vector potentials, and the extra terms that the curl produces are carried, not dropped.

Next: corrections and completion.