Kolmogorov spectrum

The Kolmogorov spectrum describes how kinetic energy is distributed across spatial scales in fully developed turbulence. For wavenumbers k in the inertial range — between the large driving scale and the small viscous dissipation scale — the energy density per unit wavenumber scales as E(k) ∝ ε^(2/3) k^(−5/3), where ε is the rate at which energy cascades through the range.

Plotted on log-log axes, it is a straight line with slope −5/3. The scaling comes from pure dimensional analysis: energy dissipation rate ε has units of m²/s³ and wavenumber k has units of 1/m, and the only combination that gives energy per unit wavenumber is ε^(2/3)·k^(−5/3). Kolmogorov's insight was recognising that dimensional analysis alone would work if you restricted attention to the inertial range. Measured ever since, the −5/3 slope is one of the most robustly-confirmed predictions in all of classical physics.

Derived by Andrey Kolmogorov in 1941, in a sequence of three papers written while the Soviet academy was evacuated from Moscow.