Critical Reynolds Number as a Topological Phase Transition in Adaptive Fractional Hydrodynamics

We present a theoretical framework that models the laminar-turbulent transition as a topological change in the dissipative operator. The order s of the fractional Laplacian is promoted from a fixed parameter to a dynamic field, governed by a variational principle that minimizes a regularized free-energy functional. This adaptive formulation continuously interpolates between the local, viscous dissipation of the Navier-Stokes equations and the non-local, anomalous dissipation characteristic of the inertial range in Kolmogorov turbulence. From this framework, we derive an analytical expression for the critical Reynolds number, Rec, by establishing a spectral balance condition where the effective dissipative capacity of the laminar operator is saturated.