Space-local Navier–Stokes turbulence

We investigate the physical-space locality of interactions in three-dimensional incompressible turbulent flow. To that, we modify the nonlinear terms of the vorticity equation such that the vorticity field is advected and stretched by the locally induced velocity. This space-local velocity field is defined by the truncated Biot–Savart law, where only the neighboring vorticity field in a sphere of radius $R$ is integrated. We conduct direct numerical simulations of the space-local system to investigate its statistics in the inertial range. We observe a standard $E\left(k\right)\propto k^{-5/3}$ scaling of the energy spectrum associated with an energy cascade for scales smaller than the space-local domain size $k\gg R^{-1}$. This result is consistent with the assumption [Kolmogorov, Dokl. Akad. Nauk SSSR 30, 299 (1941)] made for the space locality of the nonlinear interactions. The enstrophy amplification is suppressed for larger scales $k\ll R^{-1}$, and for these scales, the system exhibits a scaling consistent with a conservative enstrophy cascade.