Similarity and the Turbulent Energy Spectrum

Spherically averaged spectral densities of quantities appearing in the turbulent energy equation are assumed to be locally similar: i.e., to depend on external parameters in a similar way when referred to local scales of flux and length. Dynamical equations are given for the influence of production (including anisotropy), viscosity, buoyancy, magnetic field, and elasticity. General solutions of these equations are obtained implicitly, and the leading term in a series expansion explicitly. This similarity is shown to be a generalization of the ideas of Kolmogorov, of the modified Onsager cascade used by Corrsin, and of the formulation of Kovasznay and Shur for the influence of viscosity and buoyancy. In the case of viscosity, the prediction of a spectral cutoff by the Kovasznay formulation is seen to be misleading, and due to the retention of only the leading term in the expansion.