Energy cascade

In continuum mechanics, an energy cascade refers to the transfer of energy from large scales of motion to the small scales – called a direct energy cascade. Or a transfer of energy from the small scales to the large scales – called an inverse energy cascade. This transfer of energy between different scales requires that the dynamics of the system is nonlinear.

This concept plays an important role in the study of well-developed turbulence. It was first expressed by Lewis F. Richardson in the 1920s. Also for wind waves, in the theory of wave turbulence, energy cascades are important.

Consider for instance turbulence generated by the air flow around a tall building: the energy-containing eddies generated by flow separation have sizes of the order of tens of meters. On the other side, dissipation by viscosity takes mainly place for eddies at the Kolmogorov microscales: this will be of the order of a millimetre for the present case. At the intermediate scales, there is neither a direct forcing of the flow nor a significant amount of viscous dissipation, but there is a net nonlinear transfer of energy from the large scales to the small scales.

This intermediate range of scales, if present, is called the inertial subrange. The dynamics at these scales is described by use of self-similarity, or by assumptions – for turbulence closure – on the statistical properties of the flow in the inertial subrange. Pioneering was the deduction by Andrey Kolmogorov, in the 1940s, of his model for the wavenumber spectrum in the turbulence inertial subrange.

The largest motions, or eddies, of turbulence contain most of the kinetic energy, whereas the smallest eddies are responsible for the viscous dissipation of turbulence kinetic energy. Kolmogorov hypothesized that when these scales are well separated, the intermediate range of length scales would be statistically isotropic and that its characteristics would depend only on the rate at which kinetic energy is dissipated.

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