A Mixed-Metric Two-Field Framework for Turbulence: Emergent Stress Anisotropy and Wall Asymptotics from a Single Scalar

In our previous work~\cite{SanchisAgudoVinuesa2025PRL}, we argued that viscous dissipation in turbulence can be understood as the macroscopic imprint of microscopic path uncertainty, and showed that a kernel variance field $s(y)$ constrained by a balance condition yields both the Kolmogorov scales and the logarithmic law of the wall from a single stochastic principle. In the present work we promote $s$ to a dynamical field $s(\bm{x},t)$ with units of kinematic viscosity and develop a two-field framework in which the velocity $\ve$ and an \emph{intermittency} (or stochastic diffusivity) field $s$ evolve in a coupled way. The effective viscosity is $\nu_{\mathrm{eff}}=\nu_0+s$, but the stress tensor is generalized to include a non-linear closure driven by the commutator of strain and rotation, $[\bm{S}, \bm{\Omega}]$, capturing emergent anisotropy. The evolution of $s$ is defined as a mixed-metric gradient flow: a Wasserstein-2 gradient flow for morphology, $\Div(s\grad s)$, combined with a local $L^2$ gradient flow driven by an objective coupling term $q$. The coupling is decomposed as $q=q_{\mathrm{prod}}-q_{\mathrm{relax}}$, where production is driven by a vortex-stretching invariant, $\mathcal{I} = \|\bm{S}\boldsymbol{\omega}\|^2$. This choice ensures that production vanishes identically in strictly two-dimensional flows. We show that, under standard assumptions of constant stress, high Reynolds number and overlap-layer scale invariance, the only scale-invariant overlap-layer solution of the mixed-metric equation is $s(y)\propto y$, which recovers the logarithmic velocity profile. Thus the same mixed-metric equation organizes both wall-resolved and wall-modeled asymptotics within a single, energetically constrained framework.