Intrinsic instantaneous coarse-to-fine recoverability in multiscale chaos: The Lorenz-96 system
Abstract
In multiscale chaotic systems, a central question for closure is how much of an unresolved fine-scale target can be inferred from a simultaneous coarse observation. For a specified observation-target pair, we quantify instantaneous recoverability under the invariant snapshot measure as the fraction of target variance explained by conditioning on the retained coarse variables. Evaluating this quantity across scales produces a scale-resolved recoverability map. Applied to the Lorenz-96 system across a range of forcings, the maps reveal a strongly nonuniform landscape: lower-wavenumber targets have little variance explained by the retained lower modes, while a finite band of higher modes exhibits partial recoverability once the retained scales reach the energetically active intermediate range. Recoverability onset is organized around a reference cutoff at which two retained modes can directly couple to the target through the quadratic nonlinearity, although its precise location depends on attractor statistics. Across the tested forcings, this qualitative onset structure persists while recoverability decreases, leaving a larger fraction of target variance unexplained in more strongly driven regimes. This definition identifies the maximal explained-variance fraction attainable by instantaneous deterministic models under squared loss; the estimated maps provide a scale-dependent diagnostic for closure modeling.
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BibTeX
@misc{xu2026intrinsic,
title = {{Intrinsic instantaneous coarse-to-fine recoverability in multiscale chaos: The Lorenz-96 system}},
author = {Zhongfeng Xu and Junfeng Chen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.08323},
eprint = {2607.08323},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.08323},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Intrinsic instantaneous coarse-to-fine recoverability in multiscale chaos: The Lorenz-96 system
AU - Zhongfeng Xu
AU - Junfeng Chen
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2607.08323
UR - https://arxiv.org/abs/2607.08323
AB - In multiscale chaotic systems, a central question for closure is how much of an unresolved fine-scale target can be inferred from a simultaneous coarse observation. For a specified observation-target pair, we quantify instantaneous recoverability under the invariant snapshot measure as the fraction of target variance explained by conditioning on the retained coarse variables. Evaluating this quantity across scales produces a scale-resolved recoverability map. Applied to the Lorenz-96 system across a range of forcings, the maps reveal a strongly nonuniform landscape: lower-wavenumber targets have little variance explained by the retained lower modes, while a finite band of higher modes exhibits partial recoverability once the retained scales reach the energetically active intermediate range. Recoverability onset is organized around a reference cutoff at which two retained modes can directly couple to the target through the quadratic nonlinearity, although its precise location depends on attractor statistics. Across the tested forcings, this qualitative onset structure persists while recoverability decreases, leaving a larger fraction of target variance unexplained in more strongly driven regimes. This definition identifies the maximal explained-variance fraction attainable by instantaneous deterministic models under squared loss; the estimated maps provide a scale-dependent diagnostic for closure modeling.
ER -