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Mixed flux-equipartition solutions of a diffusion model of nonlinear cascades

We present a parametric study of a nonlinear diffusion equation which generalises Leith's model of a turbulent cascade to an arbitrary cascade having a single conserved quantity. There are three stationary regimes depending on whether the Kolmogorov exponent is greater than, less than or equal to the equilibrium exponent. In the first regime, the large scale spectrum scales with the Kolmogorov exponent. In the second regime, the large scale spectrum scales with the equilibrium exponent so the system appears to be at equilibrium at large scales. Furthermore, in this equilibrium-like regime, the amplitude of the large-scale spectrum depends on the small scale cut-off. This is interpreted as an analogue of cascade nonlocality. In the third regime, the equilibrium spectrum acquires a logarithmic correction. An exact analysis of the self-similar, non-stationary problem shows that time-evolving cascades have direct analogues of these three regimes.