ns/pulse — grow on the shear, die by viscosity

Each pulse is a plane-wave-like oscillation whose amplitude t(v) obeys a linear ODE along the pulse coordinate v: transport, the background's shear and rotation, and viscous damping that grows as the shear tilts the wavevector (7.5). This integrates that equation — in the paper's frozen-coefficient reference form, with the actual moving-normal and frame terms kept — and overlays the closed-form envelope P(v) of (7.12). The paper proves the two agree up to constants; here you can watch them.

shear parameters

a

b_s

a = −d log Ω/d log r is the swirl's angular-velocity decay; b_s the normalised radial shear of the axial velocity (4.11). Together v_s = a + b_s²/a, and the pulse only grows if v_s > 2.

pulse geometry

u*

L_s

εk²

s(v) = u*/2 + u*·v/L_s is the radial tilt of the wavevector; it passes u* at the midpoint, where amplification and damping balance. L_s ≍ S* = ℓ² is the pulse length; the errors the paper bounds by C/S* shrink as you lengthen it.

|t(v)| from the ODE envelope P(v), (7.12) Gaussian bounds e^{−c(v−L/2)²/L}, (7.16)

The same mechanism in the simplest flow there is. In plane Couette flow U = (Sy, 0) a single Fourier mode stays a single Fourier mode, but its wavevector is carried by the shear: k_y(t) = k_y(0) − S·k_x·t. Vorticity is conserved along particles, so the velocity amplitude scales like 1/|k(t)|: a wave tilted against the shear is first stretched (|k| shrinks, energy grows) and then compressed (|k| grows, energy falls), and viscosity, which damps at the rate ν|k(t)|², finishes it off. Orr found this in 1907; Craik and Criminale showed in 1986 that it is an exact nonlinear solution, because the wave's self-interaction cancels. That cancellation is why the paper's pulses can be large.

initial tilt · k_y(0)/k_x

Positive means the crests lean against the shear. The inviscid energy gain is exactly 1 + (k_y(0)/k_x)², reached when k_y passes through zero at t* = k_y(0)/(S k_x).

viscosity · ν k_x²/S

time · S·t

ψ = a(t)·cos(k_xx + k_y(t)y), k_y(t) = k_y(0) − S k_x t, a(t)|k(t)|² = a(0)|k(0)|²·exp(−ν∫₀^t|k|²) ⟹ |u′| ∝ |k(0)|²/|k(t)| · e^{−ν∫|k|²}

Why does the pulse grow at all? Displace a ring of fluid outward in a swirling flow. It keeps its angular momentum Γ = r·u_θ (no torque), so at its new radius it spins faster than its neighbours if the ambient Γ decreases outward — and a swirl surplus means a centrifugal surplus, which pushes it further out. Rayleigh (1917): unstable iff d(r·u_θ)²/dr < 0. Writing the angular velocity as Ω ∝ r^−a, that is a > 2. The paper's growth rate λ₀² = 2aF₀²(1 − 2/v_s) with v_s = a + b_s²/a is exactly this, with the axial shear b_s folded in — the centrifugal-instability criteria of Leibovich–Stewartson and Billant–Gallaire the paper cites.

angular-velocity decay · a (Ω ∝ r^−a)

axial shear · b_s

ring displacement · δr/r

ambient angular momentum Γ(r) = r·u_θ the displaced ring keeps its Γ

§2.2 · §3.3 · §7.1–7.2 · Lemmas 7.1, 7.4 · Proposition 7.2 of the paper

What a pulse is

A pulse is a spatially oscillatory velocity, localised in radius, height and time but wrapped around a complete ring, seeded by an exponentially small force and then grown by the background shear. Frozen at a point it looks like w = a·cos(ξ·x + φ) with a ⊥ ξ: a transverse wave. In the paper the phase is

Φ = pθ + p_zZ/ε + x₀R − v·(pF + p_zG) (7.3)

with integer angular frequency kp (so the wave is single-valued around the ring and has zero angular mean), carrier k = ⌈ε^−1/2⌉ so that εk² stays of order one and viscosity sits in the leading balance, and a radial component x₀ − v(pF_R + p_zG_R) that the background shear changes linearly along the pulse. F is the base angular velocity, G the base axial velocity, and v the pulse coordinate — normalised time, advancing at unit speed.

The amplitude equation

t′ + 𝒦t + m²d·t + ikm·n_Φπ = −f, n_Φ·t = 0, d = εk²|n_Φ|² (7.5)
𝒦 = ⎡ 0 −2F 0 ⎤ ⎡ 2F + RF_R 0 0 ⎤ ⎡ G_R 0 0 ⎤ (rows) (7.6)

The matrix 𝒦 is the linearisation of transport by a swirling, axially sheared background about a moving frame: a radial displacement feels the shear of the swirl (2F + RF_R) and of the axial flow (G_R), and a swirl surplus feels the Coriolis-like return −2F. Eliminating the pressure with the constraint gives the projected operator 𝒜_Φ = −𝒦 + n(nᵀ𝒦 − n′ᵀ)/|n|². In a frame B adapted to the plane n_Φ^⊥, Lemma 7.1 shows

B_ℓ(𝒜_ΦB − B′) = diag(λ, −λ) + E, λ(v) = λ₀/√(1 + s(v)²), |E| ≤ C/S*, λ₀² = 2aF₀²(1 − 2/v_s) (7.10)

one growing and one decaying direction, with the growth rate falling as the wavevector tilts (s grows) and the damping d_ref = εk²B_s²(1 + s²) rising. The normalisation B_s² = λ₀/(εk²(1 + u*²)^3/2) is chosen so that the two rates cross exactly at s = u*, the midpoint of the pulse (7.2).

The envelope

P(v) = exp ∫_{L_s/2}^v (λ(w) − d_ref(w)) dw (7.12), e^{−C(v−L_s/2)²/L_s} ≤ P(v) ≤ e^{−c(v−L_s/2)²/L_s} ≤ 1 (7.16)

Because the net rate λ − d_ref decreases at a rate between c/L_s and C/L_s and vanishes at the midpoint, log P is concave with a maximum there and the envelope is Gaussian on both sides. Lemma 7.4 then shows the actual homogeneous solution, started in the growing direction, has radial component between cP and CP throughout, with the tangential-to-radial ratio y/x = c₀√(1 + s²) + O(1/S*). The time cutoff ψ, equal to one on the middle three fifths of the pulse, acts only where P is exponentially small — so the cutoff errors and all their derivatives are flat at the singular time. That, rather than any single estimate, is why the construction can be summed.

The first tab integrates t′ = 𝒜_Φt − d·t with RK4 from t(0) = P(0)·(growing eigenvector) using the frozen reference coefficients, keeps the moving-normal term n′ and the frame drift B′ that the paper bounds by C/S*, and reports the measured growth rate against λ₀ and the measured y/x against c₀√(1 + s²). Lengthen L_s and both converge, as the estimates say they should.

Orr, and why the wave can be big

A Fourier mode in a uniform shear is carried to a new wavevector k(t) = (k_x, k_y(0) − S k_x t) without changing shape, and 2D vorticity is conserved along particles, so the streamfunction amplitude obeys a(t)|k(t)|² = const. The kinetic energy therefore goes like |k(0)|⁴/|k(t)|², peaking when the crests are vertical. With viscosity the vorticity amplitude also decays by exp(−ν∫|k|²), which is small at first and then large: growth, then decay, with a Gaussian-ish tail — the same shape as P(v). Craik and Criminale (1986) observed that such a wave on an affine background is an exact solution of the full nonlinear equations: (w·∇)w for a single transverse plane wave is a gradient and is absorbed into the pressure. The paper's pulses inherit this cancellation locally; what is left over — the interactions between different pulses, harmonics, and the slow variation of everything — is what the correction cycle of §9 mops up.

Rayleigh's criterion, in the paper's variables

The paper's shear vector is s = (a, −b_s) with a = 1 − 2D_Xlog E and b_s = 2D_XU/E (4.11). Since D_X = X∂_X = ½ r∂_r and E = √(2X)·F with F the profile of the angular velocity, a = −r∂_rlog F = −d log Ω/d log r: a swirl whose angular velocity falls like r^−a. Its angular momentum Γ = r²Ω falls like r^2−a, so Rayleigh's condition d(Γ²)/dr < 0 is a > 2, and his local growth rate is −Φ = −(1/r³)d(Γ²)/dr = 2(a − 2)Ω². Set b_s = 0 in the paper's formula: v_s = a and λ₀² = 2aF₀²(1 − 2/a) = 2(a − 2)F₀². The same number. With axial shear the effective v_s = a + b_s²/a is larger — axial shear also feeds the pulse — which is why the profile needs a nonzero radial shear of u_z near the middle plane where the rotational mechanism weakens (§2.1), and why the axis profile is given its small offset j₀ (see core → meridional flow).

The strict inequality v_s > 2 on the whole closed annulus is part (iii) of Theorem 4.6, and it is also the extra inequality that turns the "relaxed" stress-cone condition into the "admissible" one (Lemma 4.5): the cone the waves can realise is only defined where they grow (see stress).