Spike 1 (technical), The 2D Boussinesq velocity operator on a stretched grid

Step A of the 2D dynamic-rescaling port: grounding, the discretization, and its known-answer validation

Technical companion to TECHNICAL_SPIKE0_RESCALING.md (the 1D result this ports from) and the planning/derivation record ../PHASE2_SPIKE1_NOTES.md. Status: Step A built and validated; Steps B–C (the rescaled solver + the profile gate) are still ahead. Figure figures/fig9_spike1_stepA_velocity.png + committed data rebuild with python writeup/3_spikes/spike1_stepA_evidence.py. Source papers in Papers/ (gitignored).

1. Where this sits

Spike 0 validated dynamic self-similar rescaling in 1D (CLM), against a closed-form answer. Spike 1 ports the machinery to 2D Boussinesq in the Hou–Luo geometry and aims to reproduce the published Chen–Hou self-similar profile as the validation gate. The user chose the de-risked build order: build and validate the highest-risk new piece, the 2D velocity operator, as a standalone unit against a known answer before wiring the full rescaled solver. This document is that step (Step A).

The velocity operator is the 2D analogue of Spike-0's non-FFT line Hilbert transform: on the uniform periodic grid it is a trivial FFT multiplier, but the self-similar profile lives on the whole quarter-plane with a slow r^{-1/3} far-field tail, so it must be solved on a stretched grid where the FFT cannot go.

2. The grounded formulation (Chen–Hou, transcribed with equation refs)

From Chen–Hou Part I [CH22a-I, arXiv:2210.07191] §2, §7 and the MMS rigorous-numerics paper [CH25-MMS] §2 (PDFs in Papers/).

Physical 2D Boussinesq, upper half-plane ℝ²₊ = {y ≥ 0}, singularity at the origin on the wall (2.3)–(2.5):

ω_t + u·∇ω = θ_x,   θ_t + u·∇θ = 0
−Δφ = ω,   u = −φ_y,  v = φ_x,   φ(x,0) = 0     (Biot–Savart via the stream function)

with ω odd in x, θ even in x. Then φ is odd in x too, so φ = 0 on the axis x=0 as well as the wall y=0: the effective domain is the first quadrant [0,∞)² with the singular corner at the origin (the Hou–Luo geometry), and φ carries Dirichlet data on both β=0 and β=π/2.

Dynamic rescaling (one-scale; MMS (2.9)–(2.11)):

ω_τ + (c_l x + u)·∇ω = θ_x + c_ω ω
θ_τ + (c_l x + u)·∇θ = c_θ θ,          c_θ = c_l + 2 c_ω
u = ∇^⊥(−Δ)⁻¹ ω,   modulation:  c_l = 2 θ_xx(0)/ω_x(0),  c_ω = ½ c_l + u_x(0)

A self-similar profile is a steady state. Note this smooth-data scheme is one-scale (a single isotropic c_l); the "two-scale" variant cited in earlier planning was from Chen–Hou's C^{1,α}-boundary work, a different construction. So Spike-0's headline (one-scale rescaling can be stable/attracting) is the relevant precedent.

Far-field (Part I (7.1)): ω ~ g₁(β) r^α, θ ~ g₂(β) r^{1+2α}, α = c_ω/c_l ≈ −1/3. The Spike-1 gate values: c̄_l/c̄_ω ≈ −2.92 ⟺ α ≈ −0.34, c̄_ω < 0. The slow r^{-1/3} decay is the 2D analogue of Spike-0's 1/X whole-line tail: exactly why a stretched grid is mandatory.

Honest POC scope. Chen–Hou's full method (6th–8th-order B-spline FEM Poisson, adaptive mesh to 10¹⁵, a semi-analytic far-field split, 10⁻⁷ residual, INTLAB interval bounds) is a months-scale expert program, most of it built for the proof. The plan pre-committed Spike 1 to qualitative fidelity. Step A therefore uses a tractable, validated Poisson discretization in place of B-spline FEM: legitimate because the elliptic solve is textbook (contrast the exotic singular-integral line Hilbert, which had to be transcribed). The mathematics above is kept exactly.

3. The discretization (the elegant part)

Coordinates: polar (r, β), β ∈ [0, π/2], with a log-radial grid r = e^ρ (uniform ρ) and the Dirichlet angular sine basis sin(2nβ), n = 1..M−1, a DST-I with kernel sin(πjn/M) (β_j = (π/2)j/M), so the angular transform and its inverse are a pair of dense matrix multiplies (S² = (M/2)I).

The payoff is a clean cancellation. The polar Laplacian is Δφ = φ_rr + r⁻¹φ_r + r⁻²φ_ββ. Expand φ = Σ_n φ_n(r) sin(2nβ), ω = Σ_n ω_n(r) sin(2nβ). On the log grid (r_ρ = r_ρρ = r), the radial operator collapses:

φ_rr + r⁻¹φ_r = r⁻²(φ_ρρ − φ_ρ) + r⁻²φ_ρ = r⁻² φ_ρρ,

so −Δφ = ω decouples, per mode, into a constant-coefficient tridiagonal ODE:

φ_n''(ρ) − (2n)² φ_n = −r² ω_n(r).

No sparse linear algebra (there is no scipy in the venv): each mode is a Thomas solve. The homogeneous solutions r^{±2n} = e^{±2nρ} carry the physics of the two boundaries:

• near-origin regularity: φ_n ~ r^{+2n} (Robin φ_ρ = 2n φ at ρ_min);
• far-field decay: φ_n ~ r^{−2n} (Robin φ_ρ = −2n φ at ρ_max), the analytic r^{-1/3}-tail lever at POC level (the paper's semi-analytic split is the high-fidelity version of this same idea).

Velocity is recovered by the polar chain rule φ_x = cos β·φ_r − (sin β/r)·φ_β, φ_y = sin β·φ_r + (cos β/r)·φ_β, with φ_r = φ_ρ/r (central in ρ) and φ_β from the sine-basis angular derivative φ_β = Σ_n φ_n·(2n)cos(2nβ).

The origin read u_x(0). The modulation (2.11) needs u_x(0) = −φ_xy(0). Near the origin the n=1 mode dominates: φ ≈ c₁ r² sin(2β) = 2c₁ xy, so u = −φ_y ≈ −2c₁ x and u_x(0) = −2c₁ with c₁ = lim_{r→0} φ₁(r)/r². We read it by linearly extrapolating φ₁(r)/r² to r=0 over a small-r window (skipping the innermost first-order-BC nodes). This is a clean, mode-localized read: the antidote to the Spike-0 recon lesson that raw high-order pointwise origin derivatives are noise amplifiers.

Code: solver/boussinesq_velocity.py (PolarGrid, poisson_solve, velocity_from_vorticity, u_x_at_origin).

4. Validation against a manufactured known answer

Pre-committed predicate (../PHASE2_SPIKE1_NOTES.md §3): recover a manufactured (ω, u, v) to a stated tol; error must decrease under refinement; u_x(0) must match analytics. Manufactured field:

φ* = r² e^{−r} sin(2β) + r⁴ e^{−r} sin(4β)
ω* = (5r − r²) e^{−r} sin(2β) + (9r³ − r⁴) e^{−r} sin(4β)     (= −Δφ*, verified by hand)
u* = −φ*_y,  v* = φ*_x   (closed form);   u_x(0) = −2

Results (test_boussinesq_velocity.py, 5/5; figure figures/fig9_spike1_stepA_velocity.png):

The convergence panel is the key honesty check: the error is not a lucky single-grid hit, it is a clean second-order line, so the operator is converging to the right answer.

5. Honest scope

This validates the crux 2D operator; it is not a blow-up, a profile, or a proof. Step A reproduces a manufactured answer: machinery validation, by construction. Steps B (the rescaled RHS + modulation ODEs) and C (relaxing to the Chen–Hou profile: the actual gate) are still ahead, and even a flawless Step C reproduces a proven, published 2D-Boussinesq result across Wall C (2D Boussinesq ≠ 3D NS). Reproduction validates our machinery; it is not novel and not a proof. See ../PHASE2_SPIKE1_NOTES.md §4.

6. References

• [CH22a-I] J. Chen, T. Y. Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data I: Analysis, arXiv:2210.07191. (Papers/)
• [CH25-MMS] J. Chen, T. Y. Hou, … II: Rigorous Numerics, Multiscale Model. Simul. 23(1) (2025) / arXiv:2305.05660. (Papers/)
• [CH22a] J. Chen, T. Y. Hou, Finite time blowup of 2D Boussinesq and 3D Euler equations with C^{1,α} velocity and boundary, arXiv:1910.00173.