Spike 1 Step B, the rescaled 2D Boussinesq solver (technical)

Status (2026-07-25): ASSEMBLED + VALIDATED at the piece level. NOT yet shown to reproduce the profile, that is Step C (the gate), in progress. This note documents the machine: the full dynamic-rescaling right-hand side for 2D Boussinesq in the Hou–Luo geometry, the data-driven choice of evolved variables, and the piece-by-piece known-answer validation. Evidence figure + committed data: python writeup/3_spikes/spike1_stepB_evidence.py (writeup/figures/fig10_spike1_stepB_rescaled.png, writeup/data/spike1_stepB_rescaled.json).

1. What Step B builds

Step A built the velocity operator u = ∇^⊥(−Δ)⁻¹ω on the stretched quarter-plane grid. Step B wires it into the full rescaled system whose steady state is a self-similar blow-up profile (Chen–Hou, Stable nearly self-similar blowup of the 2D Boussinesq and 3D Euler equations with smooth data, arXiv:2210.07191, eqs (2.10)/(2.28)):

ω_τ = −(c_l x + u)·∇ω + η + c_ω ω
η_τ = −(c_l x + u)·∇η + (2c_ω − u_x) η − v_x ξ
ξ_τ = −(c_l x + u)·∇ξ + (2c_ω + u_x) ξ − u_y η          (v_y = −u_x used)
c_l = 2 η_x(0)/ω_x(0),   c_ω = ½ c_l + u_x(0),   c_θ = c_l + 2 c_ω

with u = ∇^⊥(−Δ)⁻¹ω from Step A. The gate target is Chen–Hou's reported profile constants (2.23): c̄_l ≈ 3.006499, c̄_ω ≈ −1.029425, ū_x(0) ≈ −2.532674, and the far-field exponent α = c̄_ω/c̄_l ≈ −0.3424 (ω ∼ r^α, θ ∼ r^{1+2α}).

2. Which variables to evolve: decided by experiment, not taste

The physical system is (ω, θ), but the paper's numerics evolve the derivatives (ω, η=θ_x, ξ=θ_y). Why it matters: the whole scheme's stability rides on the modulation origin reads ω_x(0), θ_xx(0), u_x(0), and how well those reads condition depends on the variable choice. The Spike-0 discipline says pointwise high-derivative origin reads are noise amplifiers, so we settled the choice with a measurement (experiments/spike1_stepB_decide_formulation.py):

• The angular bases follow from parity. ω, η are odd in x → they live in {cosβ, cos3β,…} (zero at the axis β=π/2, free at the wall β=0). θ, ξ are even in x → {1, cos2β,…}.
• Reading θ_xx(0) off primitive θ means an r²-curvature of two even modes, and the cos2β mode carries only θ_xx − θ_yy, contaminated by θ_yy; recovering θ_xx needs a second (constant-mode) read plus a θ(0,0) subtraction.
• Reading θ_xx(0) = η_x(0) off η is a single clean linear r-slope of η's one odd cosβ mode: the same read class as ω_x(0) and the Step-A u_x(0)=−2.0008.

Measured (fig10 panel A): the η-slope read is ~2× more accurate at every resolution and ~2× more noise-robust (mean read error under 10⁻³ grid-scale noise: 3.1% vs 6.3%). Primitive-θ was ruled out. The user chose the full 3-field variant (keep the v_x ξ coupling, don't drop it) so the steady state is exactly the Chen–Hou profile, a faithful gate.

3. The pieces, each validated against a manufactured known answer

Built de-risked, crux-piece-first (the Step-A discipline). solver/boussinesq_rescaled.py; suites test_boussinesq_transport.py (5/5) and test_boussinesq_rescaled.py (7/7).

Piece 1: the 2D upwind transport (c_l x+u)·∇f. On the log-radial × angular grid a general advection decomposes as s_ρ f_ρ + s_β f_β with

s_ρ = c_l + (u cosβ + v sinβ)/r,      s_β = (v cosβ − u sinβ)/r,

upwinded independently in ρ and β by the signs of s_ρ, s_β using Spike-0's 3rd-order Shu stencil, generalized to 2D. Manufactured advection (dilation + rotation): rel L∞ ~9.5e-6, convergence order ~2.98 (fig10 panel B), with exact structural checks (rigid rotation of a radial field → 0; pure dilation → f_ρ). Crucially the Spike-0 CFL cure carries to 2D: as r→∞, s_ρ → c_l (bounded outward dilation) and s_β → 0, so the timestep scales with Δρ independent of the domain reach, the reason a stretched grid works where a uniform one dies.

Piece 2: the velocity-gradient fields u_x, v_x, u_y (for the η/ξ reaction terms), via a polar chain-rule gradient grad_xy with central finite differences (basis-agnostic, since u=−φ_y, v=φ_x are not a pure sine/cosine series): rel L∞ ~3.9e-4, order ~1.97.

Piece 3: the origin slope read + modulation. g_x(0) for an odd field is the linear r-slope of its cosβ mode. The design payoff (fig10 panel C): c_l = 2η_x(0)/ω_x(0) is a ratio of two same-basis slope reads, so the projection quadrature bias cancels, c_l is recovered to ~3e-16 even though each slope alone carries ~2e-5. The quantity the scheme's stability most depends on is the best-conditioned quantity in it.

Piece 4: the coupled SSPRK3 integrator (RescaledBoussinesq). The RHS wiring (signs, coefficients, which field enters each reaction term) is locked by a term-by-term re-assembly test against the separately-validated sub-operators; the machine steps end-to-end with a per-step advective-CFL dt and stays finite.

4. Honest scope

This is the machine, validated to run. It is not yet shown to reproduce the Chen–Hou profile: that is Step C (the gate). Preliminary Step-C relaxation (exploratory) shows the gauge-invariant far-field exponent settling near α ≈ −0.34 (matching Chen–Hou) while the individual gauge parameters c_l, c_ω drift and the residual plateaus around 3e-2, i.e. the shape looks right but the run does not cleanly converge to steady, a drift to be diagnosed before any gate claim. And the standing caveat holds regardless of outcome: a flawless Step C reproduces a proven result (Chen–Hou 2022) across Wall C; it validates our machinery, it is not novel and not a proof. Overall Clay odds remain ~0.05%; the lottery ticket lives past this solver, in profile construction.

5. Reproduce

python test_boussinesq_transport.py          # piece 1        (5/5)
python test_boussinesq_rescaled.py           # pieces 2–4     (7/7)
python experiments/spike1_stepB_decide_formulation.py   # the formulation decision experiment
python writeup/3_spikes/spike1_stepB_evidence.py --generate       # rebuild committed data + figure