gem-lte-core

The stellarator part: wavenumber > 0 (2026-10-22)

Companion to qbo_switching_plasma.md (the QBO as the reversed-field-pinch modification of the tokamak). Code: enso_stellarator.py, qbo_low_stellarator.py; figures enso_stellarator.png, qbo_low_stellarator.png.

Mathematical form

On a common manifold F(t) with base winding k:

ζ(λ, t) = Σ_h Σ_n B_hn · Y_n(λ) · sin(2π·h·k·F(t) + φ_hn)

• Tokamak / reversed-field pinch (axisymmetric): n = 0 only; the zonal-mean flow (the QBO) carries all windings.
• Stellarator: n > 0 modes Y_n(λ) appear, and each winding h may select its own modes (a matrix B_hn that is not rank 1). Y_n is cos(nπx/L) in a bounded basin (standing) and e^{inλ} around the full equatorial ring (helical).

1. ENSO: winding-selective modes do not help

Eight equatorial Kaplan quads, 130°E–90°W, fitted 1950–2005, scored 2006–2022, common manifold (base 0.2017), windings 1–9× and the slow winding.

Structure	Parameters	Held-out mean r	Held-out pattern r	Training mean r
one shared pattern for all windings (rank 1)	28	+0.31	+0.15	+0.42
free profile per winding	160	+0.27	+0.11	+0.50
basin modes n = 0, 1, 2 mixed per winding	60	+0.24	+0.02	+0.47
one basin mode per winding	20	+0.17	+0.10	+0.38

Within the Pacific basin, every winding shares one standing pattern. The winding-specific shapes (the eastern vs central-Pacific split in enso_spatial_mode.py) fit the training years but do not predict. In the basin, ENSO acts as one mode.

2. Equatorial winds: wavenumber > 0 emerges below 70 hPa, and it follows the ENSO manifold

ERA5 monthly zonal wind, 5°S–5°N at each longitude, 1979–2025, monthly climatology removed per longitude.

Share of variance by zonal wavenumber:

hPa	1–10	20–50	70	100	125–150	200
n = 0	0.90–0.99	0.99–1.00	0.92	0.23	0.17	0.20
n = 1	0.01–0.07	≤ 0.01	0.04	0.34	0.46–0.48	0.40
n = 2	≤ 0.02	0.00	0.02	0.24	0.20–0.21	0.22

The transition is abrupt: from 70 to 100 hPa the zonal-mean share falls from 0.92 to 0.23.

Which manifold drives each wavenumber (regression on nine windings of each manifold, fitted 1979–2010; held-out correlation 2011–2022; in brackets the training fit’s percentile among 300 phase-randomized surrogates):

Level	n = 0, QBO windings	n = 0, ENSO windings	n = 1–2, QBO windings	n = 1–2, ENSO windings
5 hPa	+0.46 (100)	−0.01 (36)	+0.08	−0.05
30 hPa	+0.67 (100)	−0.01 (70)	−0.06	−0.09
50 hPa	+0.34 (83)	+0.26 (2)	−0.02	+0.03
100 hPa	−0.21 (41)	+0.10 (76)	−0.02	+0.19 to +0.22 (68–98)
125–150 hPa	−0.02 to −0.09	+0.03 to +0.05	0.00 to +0.02	+0.24 to +0.26 (79–94)
200 hPa	0.00	+0.09	−0.01	+0.25 to +0.28 (80–87)

• Above 70 hPa: the flow is axisymmetric (n = 0), and its time history follows the QBO manifold, out of sample (+0.67 at 30 hPa). The ENSO windings do not explain it.
• Below 70 hPa: wavenumbers 1–2 carry most of the variance, and they follow the ENSO manifold out of sample (+0.19 to +0.28 at every level from 100 to 200 hPa); the QBO windings give about zero there.
• So the vertical stack separates cleanly: a tokamak / reversed-field-pinch layer (n = 0, QBO manifold) over a stellarator layer (n > 0, ENSO manifold), with the boundary between 70 and 100 hPa, at the tropical tropopause.

Caveats

• The n > 0 result says that the longitudinally varying winds near the tropopause are ENSO-driven, which is already well known (the Walker circulation). What is new is only the framing and that the tidally forced ENSO windings predict it on held-out years. Held-out correlations of 0.2–0.3 are modest, and the levels are not independent.
• Surrogate percentiles for n > 0 (68–98) are suggestive, not strong.
• Monthly data cannot see the fast waves (Kelvin and mixed Rossby-gravity waves, days to weeks) that also carry n > 0 in the lower stratosphere; daily ERA5 at several longitudes would.
• In the Pacific SST itself, the stellarator-style per-winding selectivity was not supported (section 1); the n > 0 structure there is one fixed standing pattern, not winding-selective.