Explanation of the manuscript's Section 7: oscillatory realization and correction of the residual stress (pages 73 to 88)

Section 7, explained: Oscillatory realization and correction of the residual stress (pages 73 to 88)

The problem this section is handed

Sections 4 and 5 leave an axisymmetric background (uB, pB), concentrating at the origin with the prescribed growth, whose residual is minus the cylindrical divergence of an annular stress (the radial fluxes of azimuthal and of axial momentum) plus a remainder flat at the singular point [pp. 11, 73]. That residual is unbounded as t approaches 1, so it cannot be left in the force [pp. 3, 5]. The leading stress is q^{-A-1/2}T0, nonzero exactly on the annulus Xa < X < Xb [pp. 9, 84], and Section 4 placed its direction strictly inside a cone set by the local shear, with a margin that holds up to the annulus edges [pp. 31 to 33]. Section 6 built the stage: dyadic bands (Q = 2^{-ℓ}, ε = Q^h, S∗ = ℓ^2), slow boxes with a squared partition of unity, and an auxiliary torus variable Y, evaluated along a fixed map of (r, t), in which labels with overlapping slow supports get disjoint rectangles; on each rectangle a pulse coordinate v runs at unit speed in normalized time and is constant under spatial derivatives [pp. 62 to 66].

Section 7 must build a divergence-free oscillation w whose averaged quadratic products equal the leading stress and whose linear evolution in the background is solved at principal order, so that in the exact identity R(uB + w) = R(uB) + L(w) + div(w ⊗ w) only smaller terms survive [pp. 11, 73]. Sections 8 and 9 will also need an inverse for any later angular harmonic, a way to change the averaged stress by a signed amount, and exact incompressibility with controlled remainders, all on one domain at every stage [pp. 14, 88, 112].

What a person would try first, and why it fails

Stationary high-frequency waves, with the linear evolution treated as an error. This is the convex-integration reflex; the paper cites Daneri and Székelyhidi for realizing stresses by oscillations [p. 2]. It fails on scales. In chart units a wave of the needed size, √ε, meets three orders: advection of its phase at order k√ε, about one; evolution of its amplitude under shear, frame rotation and viscosity at order √ε; and only then the stress divergence it should cancel, at order ε (our reading of pp. 63, 74, 82). A stationary wave leaves the first two standing. Absorbing them into a new stress gains nothing: at frequency ε^{-1/2}, an oscillatory error of order √ε becomes a stress of order ε, as large as the original (our reading). A higher frequency is damped faster than the shear amplifies it, and a force able to sustain it would be unbounded at the singular point (our reading); the paper seeds each pulse with an exponentially small force [p. 6].

One unstable normal mode at fixed amplitude. The annulus amplifies disturbances [p. 6], so take its growing mode. First, fluid at different radii carries the crests at different speeds, so the radial wavenumber changes steadily and no wavevector stays fixed [pp. 6, 12]. Second, a growing mode cannot be held at the amplitude that gives the right stress without a force to cap it (our reading). Third, one mode has one polarization, hence one stress direction, while T0 has two independent components [p. 74].

For later corrections, square roots of the updated target. Correction stresses carry either sign and any size [p. 84]. The updated target can leave the cone, square roots are not smooth where coefficients vanish, and changing denominators could shrink the domain stage by stage (our reading of p. 112). The paper makes the point when it avoids a square root of the current covariance [p. 109].

The idea, motivated

The first attempt shows that a wave must solve its own linearized equation, at phase and amplitude order, before its stress is even visible. The second shows what such a solution can be: the background is unstable, so the fluid amplifies a disturbance from almost nothing, and an almost-nothing seed is a legal force [p. 6]. What fails is asking a growing wave to stand still. So each wave will live briefly, grow and die, and its stress will be an average over its life.

Riding the flow. A phase placed in the column is carried by it, so Φ in (7.3) contains -v(pF + pzG), which cancels advection by the rotation F and the axial flow G up to a defect of order εS∗^C (M7.2) [pp. 74 to 76]. The angular wavenumber kp is rounded to a nonzero integer, so every harmonic has zero angular mean and cos^2 averages to exactly 1/2 [pp. 74, 82]. The carrier k = ⌈ε^{-1/2}⌉ keeps εk^2 of order one [p. 74]: amplitude times wavelength, a Reynolds number for the wave, is of order one [p. 81]. Transport has an unavoidable consequence: since F and G vary in R, the radial component of the phase normal changes linearly in v (7.4), at a rate set by the tangential wavevector dotted with the shear [p. 74]. This tilt killed the normal mode; here it is scheduled. The tangential wavevector points almost along K, perpendicular to the shear direction N = g0/|g0|, with a small component along N sized so that the normalized tilt s(v) sweeps from σu∗/2 to 3σu∗/2 over a pulse of length Ls ≍ S∗ [pp. 73, 74, 76].

The amplitude equation. Leading-order incompressibility makes the amplitude orthogonal to nΦ. Pressure acts only along nΦ, and projecting it out gives the operator AΦ of (7.6), which keeps transport, the shear and frame-rotation matrix, and viscosity with both derivatives on the exponential (M7.3) [pp. 73, 75]. On the plane orthogonal to nΦ, Lemma 7.1 finds a frame in which the undamped part is diag(λ, -λ) up to O(1/S∗), with λ = λ0/√(1 + s^2) (M7.4) [p. 75]. Here λ0^2 = 2aF0^2(1 - 2/vs) is positive because the profile has vs > 2 [p. 74], the inequality Section 4 adds to its cone for these viscous waves [p. 31]. Physically this is the introduction's azimuthal surplus flung outward into slower-rotating fluid, helped by axial shear [p. 6]. Our reading, following the ledger: λ0^2 > 0 is the Leibovich and Stewartson condition, and without axial shear λ0^2 is minus Rayleigh's discriminant; the paper lists such criteria as precedents [p. 2].

Growth, then decay. Now the tilt does useful work. As |s| grows, λ falls while the damping εk^2|nΦ|^2 grows like 1 + s^2 (7.11) [p. 75]. The magnitude Bs in (7.2) makes growth and damping balance exactly where |s| = u∗, the midpoint [pp. 74, 78]. So log P vanishes with zero slope at the midpoint and has curvature of order -1/Ls: the envelope (7.12) is Gaussian of width √Ls and exponentially small in S∗ at both ends (7.16) (M7.5) [pp. 77 to 78]. This is the pulse, a free solution rising from a seed of size about e^{-cS∗} and decaying back [pp. 73, 78, 80]. Its polarization stays put: started in the growing direction, the ratio of decaying to growing coordinates solves a Riccati equation from which the scalar damping cancels, so it stays O(1/S∗) through decay as well as growth (M7.7) [pp. 80 to 81].

What a pulse can carry. Dotting the equation with the amplitude gives (7.22): energy changes by -g·(tr(tθ, tz)) minus viscous loss, and tr(tθ, tz) is exactly the radial flux of azimuthal and axial momentum (M7.8) [p. 81]. A pulse that feeds on the shear therefore carries flux with negative component along N, while the component along K is energy-neutral and must be prescribed separately [p. 81]. That is the first inequality of (7.1) (our reading). Over a whole pulse, which starts and ends near zero, the time-integrated flux must have TN < 0, since everything it dissipates came from the shear (our reading).

Two families and the cone. The auxiliary torus turns the transient into a fixed stress. Since v is normalized time along a rectangle, the Haar average at a slow point becomes an integral over the pulse coordinate, dηg = ci dv, so the covariance (7.23) is a lifetime average of the pulse's flux (7.27) [pp. 65, 82]. The Gaussian weight concentrates it where s = σu∗, giving the direction -AcN - σu∗K up to O(S∗^{-1/2}) (7.28) [p. 83]. One family is one ray; the signs σ = ± give two rays mirrored about -N, on disjoint rectangles, so no cross term appears [pp. 82 to 83]. Their positive span is TN < 0, |TK| < (u∗/Ac)(-TN) [p. 83]. As u∗ grows that opening increases toward 1/|c0|, so the second inequality of (7.1), |c0 TK/TN| < 1, says exactly that some finite tilt captures the target, and the strict margin absorbs the frame freezing and the O(S∗^{-1/2}) errors (M7.1) [pp. 74, 83]. Then y = H^{-1}T0,∗ is positive, aσ = √yσ, and W0 = √ε Σ aσbσ has covariance exactly εT0,∗, since H is by definition the covariance of the chosen pulses (M7.9) [pp. 82 to 83]. At the annulus edges yσ vanishes with the flat weight ζ, so aσ vanishes like √ζ and extends smoothly by zero [p. 83]. In physical units every power of Q cancels and the squared partition returns one copy of the target, exactly q^{-A-1/2}T0 (M7.10) [p. 84]. This is a Haar average: the angular mean at the actual Y(r, t) differs from it by a part of zero auxiliary mean, left to Section 8 [pp. 12, 88].

Corrections without square roots. Repairing the third attempt: keep aσ fixed and vary it to first order, δaσ = (H^{-1}Σ/ε)σ/(2aσ). The cross covariance is then exactly Σ for either sign, the quadratic part is retained as a remainder, and L is a right inverse of the derivative of C at W0 (M7.11) [pp. 84 to 85]. The denominator is the primary amplitude at every stage [p. 84], which lets all stages share one domain [p. 112]; Σ must not depend on the auxiliary variable [p. 84]. For nonzero harmonics of later residuals the same propagator gives Proposition 7.2: its norm is at most CP(v)/P(w), so a source carrying the envelope yields a response carrying it; because damping is scalar, higher harmonics are only more damped and never shrink the domain (M7.6) [pp. 77 to 80].

Exactness. Once amplitudes and cutoffs vary, the transverse amplitude is divergence-free only at leading order [p. 85]. The curl of the potential i nΦ × tm e^{ikmΦ}/(km|nΦ|^2) returns tm through the double cross product plus a remainder smaller by ε^{1/2-κs}, and div curl = 0 holds exactly after evaluation on the auxiliary map (M7.12) [pp. 85 to 86]. Cutting a pulse off in time leaves errors only where the envelope is e^{-cS∗}, and e^{-cℓ^2} beats every power of Q = 2^{-ℓ}, so they are flat (M7.13) [p. 87]. Corollary 7.8 gives the exact stress change when a signed increment meets the already corrected wave: Σ plus three higher-order terms, with no positivity condition (M7.14) [pp. 87 to 88].

What breaks without each move

• M7.1: once the frame is frozen, no finite tilt encloses the target with a margin, so positive squared amplitudes can fail; without λ0^2 > 0 nothing grows.
• M7.2: an untransported phase leaves an advection error larger than the wave itself (our reading); a non-integer angular frequency loses the zero angular mean; a wavenumber of another order lets viscosity either win from the start or never end the pulse.
• M7.3: the linear term of the increment identity stays at order √ε against an O(ε) target (our reading).
• M7.4: no explicit rates ±λ, hence neither the propagator bound (7.19) nor the pinned polarization.
• M7.5: without the balance at the midpoint, one end of the pulse is not exponentially small and the time cutoff error is not flat.
• M7.6: later harmonic residuals get no principal-order cancellation, and new harmonics could force smaller domains.
• M7.7: the flux direction could drift during decay, so the columns of H lose their explicit form.
• M7.8: no proof uses it (Figure 5 cites it for the amplification [p. 13]); only the reason the cone opens toward -N is lost.
• M7.9: the leading stress is not realized; one family reaches only one ray of a two-dimensional target.
• M7.10: the band sum leaves factors of Q, double counting, or cross terms instead of exactly q^{-A-1/2}T0.
• M7.11: signed corrections need square roots of a stress that may leave the cone, with denominators changing each stage.
• M7.12: waves with varying amplitudes are not solenoidal, and the increment identity, stated for divergence-free increments [p. 11], does not apply.
• M7.13: pulses cannot be truncated in time without non-flat errors reaching the force.
• M7.14: once earlier corrections and curl remainders are present, the iteration has no exact accounting of the stress change.

Three questions the section leaves open

1. Is (7.1) everything free pulses can carry? A pulse pinned to the growing eigenvector has |TK| < (-TN)/|c0| at every tilt, and the union over u∗ of the two-family cones is exactly (7.1) (our reading of pp. 74, 83), while TN < 0 is necessary by the energy balance (our reading). Can a free pulse whose polarization at its peak is not pinned reach the region in between, and at what cost in growth? It matters because the cone is a real cost upstream: Section 4 and Appendix C shape the profile to satisfy it [pp. 10, 31].

2. How deep must the bands be? The threshold q∗ comes from finitely many unquantified smallness conditions [pp. 77, 82]. Even the one displayed, S∗^2(ε + ε^2 + k^{-1}) ≤ 1 [p. 77], with h just below 1/100 and unit constants, needs ℓ above about 10,700, so q below about 2^{-10700} (our arithmetic). That is a sufficient condition of the proof, not a measured limit of the mechanism. What explicit threshold works? It matters because only an explicit, moderate one would let a pulse be exhibited at a concrete scale.

3. How sensitive is a pulse to its seed? Each pulse grows by about e^{cS∗} from a seed of that smallness [pp. 6, 78]. Which perturbations of the force or of the flow leave the realized covariance intact, given that a disturbance projecting onto the growing direction at comparable size would change it? It matters as a first step toward any stability statement about this forced construction.

Page tags

pp. 2, 3, 5, 6, 9, 10, 11, 12, 13, 14, 31, 32, 33, 62, 63, 64, 65, 66, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 109, 112.

Extraction: pages 74 to 76, 81, 83, 86 and 87 flatten scripts, matrices, bars and accents; every formula used was recoverable.