Is the Scale-Locality of the Kinetic Energy Cascade Violated by Shocks?

It is commonly believed that shocks and other discontinuities cause non-local transfer of kinetic energy (KE) —with energy at inertial scales dissipating directly into heat at molecular scales, without undergoing an inertial, scale-local cascade. This view stems from an unrefined intuition that a discontinuity, even in one dimension, is a small-scale structure. In fact, a discontinuity is quintessentially multi-scale as can be inferred from its broadband spectrum; indeed, a discontinuity requires smooth regions on either side to exist. Using a coarse-graining approach, we analyze a 1D normal shocks in compressible Navier–Stokes and in Burgers. Our analysis supports rigorous results showing that KE transfer remains scale-local and consistent with a traditional cascade. Thus, the presence of shocks alone does not imply spectrally non-local transfer. Our findings have direct implications for LES modeling of compressible turbulence in shock-dominated regimes.