A Symplectic Theory of Turbulence Closure: Hidden Reservoir Dynamics, Endogenous Stochastic Transport, and Kraichnan Dual Cascades

Abstract

Subgrid-scale parameterization of 2D turbulence must preserve the geometry enabling the dual cascade. Classical and data-driven closures often lack the symmetries required for online stability. We introduce the Symplectic Geometric Closure (SGC), derived from a geometric reduction of Multilayer Stochastic Models. Its core is a hidden reservoir of unresolved symplectic degrees of freedom feeding back on the resolved flow through a constrained Hamiltonian exchange. Stability is enforced geometrically. The interaction is generated by a symplectic functional $\mathcal{G}$ that preserves augmented enstrophy, yielding pullback boundedness for the Navier--Stokes--$β$ core and a compact random pullback attractor in the hyperviscous realization. The same generator produces an emergent Hamiltonian subgrid velocity whose forcing is exactly the Lie transport of resolved vorticity. Unresolved stochastic activity thus renormalizes the advecting geometry while maintaining the Hamiltonian structure of incompressible 2D motion. Eliminating the reservoir yields a finite-memory Eulerian theory. At one-loop line-renormalized order, reservoir contractions reproduce the operator architecture of Kraichnan-type Direct-Interaction Approximation theory. Crucially, the nested Jacobian vertex is compatible with Random Galilean transformations and imposes an intrinsic fourth-order infrared suppression, $\mathcal{O}(p^4)$, of uniform sweeping modes within the Eulerian interaction. With decorrelation controlled by strain-induced deformation rather than sweeping, the dressed Eulerian transfer theory self-consistently admits the classical $k^{-5/3}$ inverse-energy and $k^{-3}$ forward-enstrophy cascades. SGC thus provides an Eulerian geometric realization of Kraichnan's program and a principled foundation for geometrically constrained, stability-preserving data-driven closures.