Energy cascade

Remarks

The nonlinear term conserves total kinetic energy, so it only moves energy between scales; the force injects energy at large scales and viscosity removes it mainly at scales near $\eta$. In a stationary state the flux through each scale of the inertial range must therefore equal the dissipation rate. The dimensional estimate $\varepsilon \sim u^3 / L$, with $u$ and $L$ the speed and size of the large eddies, is independent of $\nu$; this is the observed dissipation anomaly.

Example: in homogeneous isotropic turbulence the flux is positive and constant across the inertial range, and the energy spectrum there follows the Kolmogorov law $E(k) \propto \varepsilon^{2/3} k^{-5/3}$. Non-example: in two-dimensional turbulence, enstrophy is also conserved by the nonlinear term. There energy goes mainly to larger scales, an inverse cascade, while enstrophy cascades to smaller scales, and $\Pi(k)$ is negative in the energy inertial range.