Hydrodynamic turbulence and the renormalization group

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Abstract
Renormalization-group (RNG) analysis, as it has been applied to infrared similarity ranges in hydrodynamic turbulence, corroborates some scaling exponents given by qualitative physical reasoning, but fails to make them more convincing. This is because local-in-wave-number interactions mediate most of the energy transfer, except in the case of asymptotic freedom in the infrared, and these interactions do not yield to low-order perturbation theory. Ultraviolet RNG analysis, which attempts to eliminate bands of modes at the low-wave-number end of a similarity range, meets the additional severe problem that renormalized mode amplitudes exhibit spurious decay due to phase mixing. Perturbative RNG analysis and pre-RNG renormalized perturbation theory both attempt to handle strong nonlinearity by chopping the dynamics into small pieces, but in different ways. Both procedures are limited by divergence, inaccessible analyticity properties, and lack of computability except in low orders.