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Self-similar enstrophy divergence in a shell model of isotropic turbulence

We focus on the early evolution of energy E and enstrophy Z when the dissipation grows in significance from negligible to important. By considering a sequence of viscous shell model solutions we find that both energy and dissipation are continuous functions of time in the inviscid limit. Inviscidly, Z takes only a finite time t* to diverge, where t* depends on initial conditions. For viscous solutions, Z peaks long after t*, but the inflection point for Z(t) provides an excellent approximation to t*. Near t*, all of our high Reynolds number solutions obey the formula ναdZ/dt = F(νβZ). Neither the function F nor the constants α and β depend on initial conditions. We use F to obtain the inviscid limit. The energy spectrum remains concave down on double logarithmic scales until t*. At t*, the spectrum becomes algebraic at high wavenumbers, i.e. E(k, t*) ∼ C0kα. Crucially, the spectral slope σ is steeper than −5/3. Thus, we conclude that the inviscid singularity at t* is not associated with the establishment of a semi-infinite Kolmogorov range. For viscous solutions, the −5/3 range builds gradually after t* starting from high wavenumbers, and Z peaks when the inertial range reaches the integral scale. Thus, the formation of the inertial range is a viscous process in our shell model.