Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse

It has been pointed out that the anomalous enstrophy dissipation in non-smooth weak solutions of the two-dimensional Euler equations has a clue to the emergence of the inertial range in the energy density spectrum of two-dimensional turbulence corresponding to the enstrophy cascade as the viscosity coefficient tends to zero. However, it is uncertain how non-smooth weak solutions can dissipate the enstrophy. In the present paper, we construct a weak solution of the two-dimensional Euler equations from that of the Euler-$\ensuremath{\alpha} $ equations proposed by Holm, Marsden & Ratiu (Phys. Rev. Lett., vol. 80, 1998, pp. 4173–4176) by taking the limit of $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$. To accomplish this task, we introduce the $\ensuremath{\alpha} $-point-vortex ($\ensuremath{\alpha} \mathrm{PV} $) system, whose evolution corresponds to a unique global weak solution of the two-dimensional Euler-$\ensuremath{\alpha} $ equations in the sense of distributions (Oliver & Shkoller, Commun. Part. Diff. Equ., vol. 26, 2001, pp. 295–314). Since the $\ensuremath{\alpha} \mathrm{PV} $ system is a formal regularization of the point-vortex system and it is known that, under a certain special condition, three point vortices collapse self-similarly in finite time (Kimura, J. Phys. Soc. Japan, vol. 56, 1987, pp. 2024–2030), we expect that the evolution of three $\ensuremath{\alpha} $-point vortices for the same condition converges to a singular weak solution of the Euler-$\ensuremath{\alpha} $ equations that is close to the triple collapse as $\ensuremath{\alpha} \ensuremath{\rightarrow} 0$, which is examined in the paper. As a result, we find that the three $\ensuremath{\alpha} $-point vortices collapse to a point and then expand to infinity self-similarly beyond the critical time in the limit. We also show that the Hamiltonian energy and a kinematic energy acquire a finite jump discontinuity at the critical time, but the energy dissipation rate converges to zero in the sense of distributions. On the other hand, an enstrophy variation converges to the $\delta $ measure with a negative mass, which indicates that the enstrophy dissipates in the distributional sense via the self-similar triple collapse. Moreover, even if the special condition is perturbed, we can confirm numerically the convergence to the singular self-similar evolution with the enstrophy dissipation. This indicates that the self-similar triple collapse is a robust mechanism of the anomalous enstrophy dissipation in the sense that it is observed for a certain range of the parameter region.