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According to Richardson's cascade description of turbulence, large vorticesbreak up to form smaller ones, thereby transferring kinetic energy towardssmaller scales. Energy dissipation occurs at the smallest scales due toviscosity. We study this energy cascade in a phenomenological model of vortexbreakdown. The model is a binary tree of decreasing masses connected bysoftening springs, with dampers acting on the lowest level. The masses andstiffnesses between levels change according to a power law. The differentlevels represent different scales, enabling the definition of "masswavenumbers". The eigenvalue distribution of the model exhibits a devil'sstaircase self-similarity. The energy spectrum of the model (defined as theenergy distribution among the different mass wavenumber) is derived in theasymptotic limit. A decimation procedure is applied to replace the model withan equivalent chain oscillator. We show that for a range of stiffness parameterthe energy spectrum is qualitatively similar to the Kolmogorov spectrum of 3Dhomogeneous, isotropic turbulence and find the stiffness parameter for whichthe energy spectrum has the well-known -5/3 scaling exponent.

Reproducing the Kolmogorov spectrum of turbulence with a hierarchical linear cascade model