Logit--Koopman coordinates for scale-resolved forecast-error growth in a three-level Lorenz-96 system
Abstract
Forecast-error growth in multiscale chaotic systems involves both advancement toward saturation and evolution of the error's scale distribution. We introduce Logit--Koopman coordinates to describe these processes together. For error energies normalized by fixed reference levels, logistic growth has exponential odds as exact Koopman eigenfunctions and affine logits as growth clocks, with an explicit inverse to error energy. Several band clocks define a longitudinal coordinate for shared advancement and transverse coordinates for departures from the reference band-growth relation. An exact transformed error-energy budget resolves production, conservative redistribution, interlevel exchange and dissipation into contributions to clock speed and transverse motion; an equivalent reference-law decomposition exposes the effect of ensemble heterogeneity. In uncoupled and coupled three-level Lorenz-96 regimes, 200 forecasts per regime exhibit nearly affine band-mean logits (R2=0.965--$0.993$), alongside distinct timing, residual and launch-anomaly correlations. Recovered error fields reveal net slow-to-fast interlevel error-energy exchange in the coupled control, which removes slow-level error energy at a nearly steady relative rate near 1.2 per model time unit. Fast-only initial perturbations nevertheless seed slow-level errors immediately; after this transient, their mean approximately follows the control growth curve delayed by about 0.7 model time units. The slower middle-band clock reflects the full physical balance and an error deficit maintained at the dominant wavenumbers, rather than the strongest transformed interlevel sink. The contribution is a prescribed reference geometry in which growth timing, relative scale-evolution and signed physical exchange become separately measurable.
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BibTeX
@misc{pena2026logitkoopman,
title = {{Logit--Koopman coordinates for scale-resolved forecast-error growth in a three-level Lorenz-96 system}},
author = {Malaquias Peña},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.34779},
eprint = {2610.34779},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.34779},
}
RIS
TY - UNPB
TI - Logit--Koopman coordinates for scale-resolved forecast-error growth in a three-level Lorenz-96 system
AU - Malaquias Peña
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.34779
UR - https://arxiv.org/abs/2610.34779
AB - Forecast-error growth in multiscale chaotic systems involves both advancement toward saturation and evolution of the error's scale distribution. We introduce Logit--Koopman coordinates to describe these processes together. For error energies normalized by fixed reference levels, logistic growth has exponential odds as exact Koopman eigenfunctions and affine logits as growth clocks, with an explicit inverse to error energy. Several band clocks define a longitudinal coordinate for shared advancement and transverse coordinates for departures from the reference band-growth relation. An exact transformed error-energy budget resolves production, conservative redistribution, interlevel exchange and dissipation into contributions to clock speed and transverse motion; an equivalent reference-law decomposition exposes the effect of ensemble heterogeneity. In uncoupled and coupled three-level Lorenz-96 regimes, 200 forecasts per regime exhibit nearly affine band-mean logits (R^2=0.965--$0.993$), alongside distinct timing, residual and launch-anomaly correlations. Recovered error fields reveal net slow-to-fast interlevel error-energy exchange in the coupled control, which removes slow-level error energy at a nearly steady relative rate near 1.2 per model time unit. Fast-only initial perturbations nevertheless seed slow-level errors immediately; after this transient, their mean approximately follows the control growth curve delayed by about 0.7 model time units. The slower middle-band clock reflects the full physical balance and an error deficit maintained at the dominant wavenumbers, rather than the strongest transformed interlevel sink. The contribution is a prescribed reference geometry in which growth timing, relative scale-evolution and signed physical exchange become separately measurable.
ER -