When Does Forecast-Error Energy Grow Logistically in Geophysical Turbulence?

Abstract: Coarse-graining can yield a simple macroscopic growth curve in a bounded chaotic system even when constituent scales follow different clocks. The distinction matters as reduced-order and generative models compress multiscale forecast uncertainty into learned coordinates. We ask when forecast-error energy admits a logistic law. From the exact twin-error budget and correlated and decorrelated spectra, we derive two scalar limits: an invariant decorrelation amplitude, logistic only when contributing scales share one shape and one clock, and a self-similar upscale error front whose law depends on spectral slope and front speed. With local-strain scaling, the front predicts exponential error-energy growth for the canonical barotropic-vorticity spectrum and linear growth for the surface-quasigeostrophic spectrum. Stationary forced surface-quasigeostrophic twins test the logistic admission conditions. A response-blind partition of 16 trajectories gives cluster-mean logistic root-mean-square deviations 0.080 and 0.093, although every trajectory has resolved clock heterogeneity. An exact averaging identity shows how signed shape and clock corrections cancel, producing a nearly logistic aggregate while constituent scales retain distinct clocks. Mechanism identification therefore requires more than goodness of fit: independent shape, clock, and residual tests are required. These admission conditions provide physics-based guardrails for compact representations of chaotic systems and generative forecast ensembles.