This talk is the follow-up to the studies presented in the workshop at the Fields Institute in 2016. In the workshop, we have introduced two mathematical models in order to understand the mechanism of an anomalous enstrophy cascade in terms of fluid dynamics.

The first model is based on the 2D incompressible Euler-$\alpha$ equations for the initial vorticity data with $\delta$-measure distributions, called point vortex-$\alpha$(PV$\alpha$) system. Considering three $\alpha$-point vortices, we construct a singular incompressible and inviscid flow by taking the $\alpha \rightarrow 0$ limit of their evolution. We show rigorously that, under a certain condition on the vortex strengths, the limit solution tends to a self-similar finite-time triple vortex collapse. We have also proved that the enstrophy dissipates in the sense of distributions at the event of the collapse, suggesting that the triple vortex collapse is a mechanism of an anomalous enstrophy dissipation. In the present talk, let us consider the Euler-Poincare models, a generalization of the Euler-$¥alpha$ model, in which the incompressible velocity field is dispersively regularized by a smoothing function. We provide a sufficient condition for the existence of the triple collapse dissipating the enstrophy, and we then confirm that the condition is satisfied with the Gaussian regularization and the vortex blob regularization, which are commonly used in numerical computations of the Euler equations. It indicates that the enstrophy dissipation via the collapse of three point vortices is a generic phenomenon that is not specific to the Euler-$\alpha$ model but universal within the Euler-Poincare models.

The second model is a one-dimensional hydrodynamic partial differential equation for the vorticity, which is a generalization of the Constantin-Lax-Majda equation. Any solution to the equation conserves its $L^p$ norm ($p \geq 1$) in the inviscid case, which will be a cascading quantity in the viscous case. In the previous talk, we have considered the case of $p=2$, in which the invariant quantity corresponds to the enstrophy. With a large-scale random forcing and small viscosity, we have found numerically that solutions become turbulent, and an inertial range in the energy spectra with an enstrophy cascade emerges, which is a common property shared with 2D turbulence. In the present talk, considering the wider range of $p \in [1,4]$, we show numerically that the model exhibits the cascade of the inviscid invariant, the broad energy spectrum with a correction to the dimensional-analysis prediction connected to singular steady solutions and self-similarity in the dynamical system structure, which will be reported in the present talk. The 1D model is amenable to mathematical and numerical analysis than the Navier-Stokes equations, which leads to a theoretical understanding of the emergence of inertial range.