Turbulence as a Deterministic Bifurcation Cascade and Its Cosmological Manifestation: A Unified Theory via the Gauge Enstrophy and the Inverse Kraichnan Cascade

This paper presents a unified theory of turbulence as a deterministic cascade of bifurcations driven by the inverse Kraichnan cascade of gauge enstrophy. Wereplace the classical linear stability paradigm of Orr–Sommerfeld with a nonlinear variational principle: any shear flow spontaneously organises into a three-tier structure (2D periphery → axisymmetric filament → 2D core) that maximises gauge enstrophy for fixed kinetic energy. The conservation of this enstrophy forces an inverse energy cascade, pumping energy from small to large scales. Viscosity acts not as a passive damper but as an active order parameter controlling mode selection, phase synchronisation, and the bifurcation sequence. As the Reynolds number increases, the system undergoes a Ruelle–Takens–Newhouse cascade: a fixed point(laminar flow) bifurcates to a limit cycle (K´arm´an vortex street), then to a T2 torus, and finally to a strange Lorenz attractor (fully developed turbulence). The critical Reynolds number is computed from first principles rather than fitted empirically.We then demonstrate the scale-invariance of this mechanism by applying it to cosmology. The Euler equations on a Friedmann–Lemaˆıtre–Robertson–Walker background are isomorphic to the Navier–Stokes equations in conformal coordinates.Projecting onto 2D angular modes yields a gauge enstrophy invariant that is conserved in the early Universe. The inverse cascade transfers energy from Planck-scale quantum fluctuations to cosmological scales, generating an effective cosmologicalconstant Λeff as the volume-average of the vortex stress tensor. The late-time acceleration of the Universe emerges naturally: Λeff ∝ a−4 decays faster than matter density ρm ∝ a−3, so deceleration dominates until ρm drops below Λeff at redshiftz ≈ 0.7, consistent with Supernova, BAO, and CMB observations. No fine-tuning is required.The theory unifies laboratory turbulence, atmospheric vortex dynamics, and cosmic acceleration under a single mathematical principle: the gauge enstrophycascade.