What question did this study set out to answer?

This research aims to derive Kolmogorov scaling using a variational principle and to explore anomalous dissipation in turbulent systems.

March 21, 2026Open Access

Anomalous Dissipation and the Turbulent Cascade: A Variational Free-Energy Derivation of Kolmogorov Scaling

Key Points

This research aims to derive Kolmogorov scaling using a variational principle and to explore anomalous dissipation in turbulent systems.
Utilized a variational principle to establish the Kolmogorov k−5/3 energy spectrum
Introduced the Downhill Flux Condition (DFC) for analyzing anomalous dissipation
Developed a falsification protocol for Direct Numerical Simulations (DNS)
Recast DFC into an energetic stability class
Conducted numerical verification of critical theorems regarding perturbation tests
Confirmed the Kolmogorov scaling as the unique minimizer of the scale-resolved free-energy functional
Established that anomalous dissipation occurs under DFC
Achieved exact results for theorem verifications with a perfect score in perturbation tests
Demonstrated a finite-N spectral gap through Markov structure in turbulence modeling

Abstract

A variational principle yields the Kolmogorov k−5/3 energy spectrum as the unique minimiser of a scale-resolved free-energy functional. Anomalous dissipation follows under the Downhill Flux Condition (DFC). The DFC hierarchy provides a concrete DNS falsification protocol. Published in English and German versions. Part of the Functional Stability Theory (FST) programme. Changes in Version 1. 3 (March 2026) No substantive changes (already journal-grade). Recompiled with latest cross-references. Changes in Version 1. 2 (March 2026) Critical: Fixed placement Changes in Version 1. 1 (March 2026) Major: Numerical verification of Theorem 3. 1: FEK41=0 exactly, 300/300 perturbation tests confirm strict convexity Major: DFC recast as energetic stability class Major: Finite-N spectral gap via shell-model Markov structure (GOY/SABRA)

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