Exact form of the joint probability functional ρ[φ,z]

Determine the exact functional form of the joint probability density functional ρ[φ,z] over the scalar field φ(x) and the surface z(x) in the nonplanar vortex gas model of turbulent circulation, subject to the constraint that ρ[φ,z] reproduces the two-point correlation ⟨φ(x,z) φ(x′,z′)⟩ = −(1/(4π)) ln(((x−x′)^2 + (z−z′)^2 + η^2)/L^2). The current analysis assumes a Gaussian structure ρ[φ,z] = C exp{−S[φ,z]}, but the exact form is unknown.

Background

In developing the nonplanar vortex gas model (VGM), the authors express the circulation characteristic function as a functional integral over both the surface z(x) and the scalar field φ(x), weighted by a joint probability density functional ρ[φ,z]. This functional is required to reproduce the specified log-correlated two-point function for φ, which models energy dissipation fluctuations via Gaussian multiplicative chaos.

Because the exact form of ρ[φ,z] is not known, the authors assume a Gaussian structure defined by a quadratic action S[φ,z]. This assumption underpins the saddle-point analysis that identifies minimal surfaces as optimal within the inertial range. A precise determination of ρ[φ,z] would provide a more rigorous foundation for the model and could impact predictions, especially regarding corrections to minimal surfaces and circulation statistics near the dissipation scale.