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Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse
| Title: | Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse |
| Authors: | Sakajo, Takashi Browse this author →KAKEN DB |
| Keywords: | Hamiltonian theory | | low-dimensional models | | vortex dynamics |
| Issue Date: | 10-Jul-2012 |
| Publisher: | Cambridge University Press |
| Journal Title: | Journal of Fluid Mechanics |
| Volume: | 702 |
| Start Page: | 188 |
| End Page: | 214 |
| Publisher DOI: | 10.1017/jfm.2012.172 |
| Abstract: | 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-α equations proposed by Holm, Marsden & Ratiu (Phys. Rev. Lett., vol. 80, 1998, pp. 4173-4176) by taking the limit of α → 0. To accomplish this task, we introduce the α-point-vortex (αPV) system, whose evolution corresponds to a unique global weak solution of the two-dimensional Euler-α equations in the sense of distributions (Oliver & Shkoller, Commun. Part. Diff. Equ., vol. 26, 2001, pp. 295-314). Since the αPV 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 α-point vortices for the same condition converges to a singular weak solution of the Euler-α equations that is close to the triple collapse as α → 0, which is examined in the paper. As a result, we find that the three α-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 δ 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. |
| Rights: | Copyright © Cambridge University Press 2012 |
| Type: | article |
| URI: | http://hdl.handle.net/2115/52952 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)
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Submitter: 坂上 貴之
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