Vortex dynamics of turbulent energy cascades

Employing vortex dynamics, we explore the turbulent cascade mechanisms in Schroedinger and Navier–Stokes fluids. While both cascades are driven by vortex instabilities, the ability of Navier–Stokes vortices to stretch and exhibit complex core dynamics significantly affects the resulting turbulence behavior. In dilute Schroedinger turbulence at scales smaller than the intervortex distance, Aarts-de Waele instabilities trigger reconnection-driven Kelvin wave energy cascades, transferring energy from the reconnection scale to smaller scales. At sufficiently long times, these cascades create a high-wavenumber bottleneck before transitioning into a k^-5/3 local-interaction cascade scaling regime. Energy accumulates in the length scales preceding the bottleneck, triggering partial spectrum equilibration and resulting in a positive scaling exponent there, which differs from the equilibrium value of k². At scales larger than the intervortex distance, the spectrum scales as k², which is indicative of finite linear impulse in the system. In Navier–Stokes turbulence, the self-stretching of large-core vortices triggers an energy cascade to smaller scales, which is then intensified by the stretching of emergent vortex structures created by Crow or helical vortex line instabilities. The k^-5/3 scaling arises only once this iterative process has progressed sufficiently to confine flow enstrophy within tubular regions, where the core size becomes a sufficiently small fraction of the overall system size. This confinement causes the vortices to appear quasisingular when measured on large-scale units. The scaling of the entire-system spectrum is determined by the spectrum of the quasi-singular structures at the culmination of the cascade process, rather than by the cascade process itself.