Foundations of a solved-volatility stochastic turbulence closure: It\^o--Hencky kinematics, source-consistent momentum and finite-correlation realisation

Abstract

Most stochastic closures prescribe a covariance tensor, a noise basis or an eddy-viscosity field. This paper develops a different framework in which the displacement-volatility field is solved together with the resolved velocity. The starting point is a one-channel It\^o configuration map. A local matrix-logarithm expansion gives distinct material and spatial Hencky increments, their quadratic-variation drifts and the exact pathwise volume constraint. Under constant density, one Brownian channel, pathwise isochoricity and no independent martingale in the resolved Eulerian drift, the material pull-back momentum equation is shown to be the on-shell form of the full stochastic Reynolds transport balance when the momentum-source covariation is retained. The resulting velocity--volatility--pressure system has an index-one differential--algebraic structure. Virtual power fixes the mechanical type of the stress impulse and separates work from quadratic covariation. A finite-correlation precursor then gives a Green--Kubo realisation of the solved displacement covariance. State dependence adds a Lyapunov noise-induced drift, while dynamic boundaries add reaction work and active/passive covariance compatibility conditions. The analysis also gives four limits: a non-zero Brownian transport limit cannot retain finite ordinary unresolved kinetic energy; total energy alone does not fix entropy production; wall tangency limits covariance rank rather than the number of stochastic modes; and a homogeneous decoupled Helmholtz--Stokes equation has only the trivial periodic solution. The result is a theory-complete, testable closure architecture. Developed turbulent statistics, logarithmic wall scaling and computational-fluid-dynamics validation are deliberately left to the expanded fluid-mechanics study.