Expressing turbulent kinetic energy as coarse-grained enstrophy or strain deformations

In turbulent flows, the fluid element is deformed by chaotic motion due to the formation of sharp velocity gradients, yet an exact relationship between these deformations and kinetic energy remains elusive. In the context of incompressible and homogeneous turbulence, this work derives an exact identity connecting velocity gradient norms across the scales to kinetic energy, offering a novel decomposition in terms of strain-rate deformations and vortical motion. The formulation also leads to an exact real-space representation of the kinetic energy spectrum, providing new insights into the Kolmogorov constant and spectral scaling in hydrodynamic turbulence theory and beyond.