Stationary forcing produces a non-steady radial 'pulsing'; instability worsens as dt shrinks (over-remeshing); Hs amplitude not dt-converged

[mochell]

Summary

Under a stationary, axisymmetric TC forcing the wave field should reach a steady state, but it does not: it periodically launches radial waves outward from the storm centre ("pulsing"). This is purely numerical — the forcing is constant in time and radially symmetric.

It is NOT a high-CFL instability — it grows as dt shrinks (over-remeshing)

Reducing the remesh timestep dt (which lowers cg·dt/dx) makes the pulsing dramatically worse, with a sawtooth signature (slow bleed-down then snap-back). Late-time (t>10 h) max-Hs ripple, Holland CAT4, Rm=75 km, dx=10 km, P=2:

dt	mean max Hs	ripple (rel. peak-to-peak)
20 min	24.5 m	10.2%
5 min	21.1 m	4.2% (calmest)
2 min	20.3 m	20.4%
1 min	19.0 m	37.7%

There is a sweet spot near dt=5 (moderate CFL). The failure mode is remesh frequency: when a particle moves ≪ one grid cell per step, the advect → deposit → remesh → deposit round-trip (a lossy interpolation/reconstruction every step) accumulates error → sawtooth. So "reduce dt for stability" is exactly backwards here.

Figure: pulsing_maxHs_timeseries_P2.png. Please attach from plots/hurricanes/comparison/.

Amplitude is not dt-converged

The mean max Hs drifts systematically with dt (19.0 → 20.3 → 21.1 → 24.5 m for dt = 1 → 2 → 5 → 20 min). The scheme is not converged in amplitude w.r.t. the remesh timestep.

Pre-existing — not introduced by the B-spline work (#60)

The pulsing is present at P=1 (CIC, the original scheme) too — late-time ripple at Rm75 dx10: P=1 6.4%, P=2 4.2%, P=3 8.4%. It was previously masked by the grid-axis anisotropy; once #60 made the field isotropic, the temporal oscillation is clearly visible as clean radial waves.

Resolution sensitivity

A better-resolved storm (Rm=200 km, Rm/dx=20 vs 7.5) has a much smoother Hs field — the sharp radial pulsing is strongly suppressed (hovmoller_Rm200_P2_y0.png).

Root cause (hypothesis)

The lossy deposit ↔ GetVariablesAtVertex reconstruction applied at every remesh step; more remeshes per unit time → more accumulated error. Likely coupled to the eye reconstruction singularity in #63 (the λp eye spikes correlate with these resets).

Figures (attach from plots/hurricanes/comparison/): pulsing_maxHs_timeseries_P2.png, hovmoller_Hs_P2_holland_CAT4_y0.png, hovmoller_Rm200_P2_y0.png.

[mochell]

Characterized: the pulsing is a remesh-induced amplitude limit cycle (numerical, not physical)

Investigated with a stability suite (S05–S08) on the bspline-pic-deposition branch. Stationary forcing must reach a steady state, so any sustained oscillation is a numerical artifact. Here is what it is and what controls it.

Where the periodicity lives

Single-vertex point-probe time series (S08) — at the TC eyewall (r = Rmax) and one node right of the C2 wind/calm step — show:

• wind speed and momentum direction are perfectly constant, while
• energy E and momentum magnitude |m| oscillate in phase.

So it is not a forcing effect and not a reconstruction-angle effect — it is the wave amplitude limit-cycling through the advance → deposit → remesh loop.

Two distinct triggers (S07 resolution × dt sweeps, additive deposition)

	C1 (symmetric TC)	C2 (wind/calm step)
shape	smooth, period ~5 h	sawtooth relaxation, period ~15 h
organizing number	Rmax / dx (eyewall resolution)	wave Courant c_g·dt/dx
collapses on Courant?	no (resolution curves stay separated)	yes (clean cliff)
cured by	resolving Rmax with ≳8–10 (ideally ~20) points	c_g·dt/dx ≳ 2–3 (≈ U_max·dt/dx ≳ 10–12)
dt behaviour	sweet spot ~20 min; too-small AND too-large dt are worse	larger dt / Courant ⇒ steadier

• C1 has an intrinsic forcing scale (Rmax = 50 km). Under-resolving the eyewall drives the cycle: cv_mean(E) drops ~10× from dx = 15 km → 5 km; a well-resolved Nx = 241 (dx = 2.5 km) largely damps the eyewall oscillation.
• C2 has no intrinsic scale (a wind discontinuity) — the pulsing is genuine over-remeshing and collapses onto the wave Courant number: above the threshold cv_mean → 1e-4…1e-7 (steady).

Note on "worsens as dt shrinks" (issue title): confirmed at fixed grid — but it is the advective/wave Courant number that organizes C2, and grid resolution that organizes C1. Shrinking dt without regard to c_g·dt/dx makes the over-remeshing pulsing worse, not better.

Approaches that did NOT fix it (deposition-side — rejected)

The pulsing is not a deposition-rule problem. Prototyped and empirically rejected:

• Gather smoothing (stencil-weighted node→particle gather): worsened steadiness, destabilised previously-steady cases. Reverted.
• B-spline order 2 vs 1: order-independent; order 2 not better (sometimes worse).
• Wind-sea merge! (Hanson & Phillips 2001 wave-age contest, favouring wind sea): no-op for unidirectional seas (C2 bit-identical), and worsens the rotating eyewall (C1). Kept only as an opt-in windsea_merge model toggle (default off) for multi-system partitioning physics — not as a pulsing fix.

Practical guidance

• Resolve the forcing gradient. For a TC, ≳8–10 (ideally ~20) grid points across Rmax (dx ≲ Rmax/10). Dominant control for eye pulsing.
• Keep the wave Courant c_g·dt/dx ≳ 2–3 (≈ U_max·dt/dx ≳ 10) near sharp forcing interfaces. Do not shrink dt below this thinking it is safer.
• dt sweet spot ≈ 20 min for these configs.
• Metric caveat: domain cv_mean(E) understates eyewall structure — a case can score "smooth" while the eyewall still oscillates (visible in the S08 point probe and the dt=40 video). Judge local smoothness with point probes / field videos, not a domain scalar.

Open / next

The remaining untested lever is reducing remesh frequency (sub-cycling advection so remeshing happens less often than every advance step), which directly targets the C2 over-remeshing relaxation cycle. Not yet implemented.

Reproduce: analysis/stability/S05_…, S06_…, S07_…, S08_… plus the writeup analysis/stability/FINDINGS_zero_momentum_and_pulsing.md. Key figures: S07_*_collapse_groupings_P2.png, S08_C1_eyewall_probe.png, S08_C2_step_probe.png, and S07_*_ref_dt{20,40}_P2.mp4.