Fractional Navier–Stokes equations from a hierarchical cascade: Derivation, exact analytical solutions, and comparison with turbulent pair-dispersion data

Fractional Navier–Stokes equations have been postulated phenomenologically for decades to model anomalous transport, with the fractional order α fitted to data. We derive these equations as the formal continuum limit of a discrete hierarchical fluid parcel system with Riemann–Liouville dynamics, and prove that the piecewise-constant parcel field converges strongly in L2(0,T;L2(Ω)) to a weak solution of the fractional Navier–Stokes system. The temporal order is fixed by the cascade branching number through αk=2−2/(Nk+1), so that Nk=1, 2, and ∞ recover the classical Navier–Stokes limit (α=1), the Richardson regime (α=4/3), and wave transport (α→2). We obtain closed-form solutions for the fractional Stokes first problem, channel start-up, and Couette flow, verified against the classical limits at machine epsilon. An analytic mapping connects the parametric N-body velocity vk(θ)=2 sin 2θ to the channel Fourier amplitudes, presented as a mathematical construction rather than a physical mechanism. The binary-cascade exponent α=4/3 reproduces the direct numerical simulation (DNS)-confirmed ⟨r2⟩∝εt3 pair-dispersion law through the Richardson closure, with constant g=0.607 inside the DNS range [0.50, 0.70]; g enters through the Kolmogorov–Obukhov constant C2 and is not a free parameter. Nine independent superdiffusive measurements from DNS, spacecraft, and laboratory plasmas match the cascade-to-α map within uncertainties under a pre-specified χ2 test, presented as a comparison with the broader fractional transport class rather than direct validation of the momentum equation.