A Symplectic Theory of Turbulence Closure: Hidden Reservoir Dynamics, Endogenous Stochastic Transport, and Kraichnan Dual Cascades
Abstract
The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and β-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier--Stokes--β realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved--reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson--Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the k−5/3 inverse-energy and k−3 forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.
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BibTeX
@misc{chekroun2026symplectic,
title = {{A Symplectic Theory of Turbulence Closure: Hidden Reservoir Dynamics, Endogenous Stochastic Transport, and Kraichnan Dual Cascades}},
author = {Mickaël D. Chekroun and James C. McWilliams},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.06606},
eprint = {2608.06606},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.06606},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Symplectic Theory of Turbulence Closure: Hidden Reservoir Dynamics, Endogenous Stochastic Transport, and Kraichnan Dual Cascades
AU - Mickaël D. Chekroun
AU - James C. McWilliams
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.06606
UR - https://arxiv.org/abs/2608.06606
AB - The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and β-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier--Stokes--β realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved--reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson--Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the k^−5/3 inverse-energy and k^−3 forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.
ER -