The K´arm´an Vortex Street as an Oscillation Between the Kraichnan and Kolmogorov Cascades: A Gauge Enstrophy Theory of Periodic Vortex Shedding

We present a rigorous mathematical theory of the K´arm´an vortex street as a self-sustained oscillation between two regimes of turbulent energy transfer. Using the gauge enstrophy formalism developed in our previous works, we prove that the periodic shedding of vortices in the wake of a bluff body is driven by an alternating dominance of the inverse Kraichnan cascade (vortex formation phase) and the direct Kolmogorov cascade (vortex decay phase). The governing parameter is the KAM winding index K = Ecal/(EK∆α), where Ecal is the gauge enstrophy of the vortex filament, EK is the kinetic energy of the wake, and ∆α is the Diophantine distance of the frequency ratio from the nearest rational resonance.We prove four main results. First, the birth of the Karman vortex street at Re ≈ 47 is a supercritical Hopf bifurcation governed by the Lorenz attractor, with the critical value Kcrit = ρc ≈ 28 inherited from the universal homoclinic explosion parameter. Second, the oscillation of K between subcritical and near-critical values corresponds to the alternation between the inverse cascade (vortex roll-up) and the direct cascade (vortex shedding and dissipation). Third, the Strouhal number St ≈ 0.21 for the cylinder and St ≈ 0.15 for the sphere are derived from firstprinciples as the Diophantine optimal approximations of the frequency ratio, explaining their remarkable universality across a wide range of Reynolds numbers. Fourth, the topology of the generated vortex is classified by the local curvature tensor of the body surface: parabolic points produce cylindrical vortices (the Karman vortex street), elliptic points produce vortex rings (the Karman vortex ring), andhyperbolic points produce helical vortices.We further prove that the transition to a turbulent wake is triggered by a three-dimensional instability of the vortex filament at K = 28, and derive the spatial decay law for the vortex street. The theory contains no adjustable parameters. Practical applications are developed in a companion paper on the boundary layer.