Abstract
Dissipation of enstrophy in 2D incompressible flows in the zero viscous limit is considered to play a significant role in the emergence of the inertial range corresponding to the forward enstrophy cascade in the energy spectrum of 2D turbulent flows. However, since smooth solutions of the 2D incompressible Euler equations conserve the enstrophy, we need to consider non-smooth inviscid and incompressible flows so that the enstrophy dissipates. Moreover, it is physically uncertain what kind of a flow evolution gives rise to such an anomalous enstrophy dissipation. In this paper, in order to acquire an insight about the singular phenomenon mathematically as well as physically, we consider a dispersive regularization of the 2D Euler equations, known as the Euler-\(\alpha \) equations, for the initial vorticity distributions whose support consists of three points, i.e., three \(\alpha \)-point vortices, and take the \(\alpha \rightarrow 0\) limit of its global solutions. We prove with mathematical rigor that, under a certain condition on their vortex strengths, the limit solution becomes a self-similar evolution collapsing to a point followed by the expansion from the collapse point to infinity for a wide range of initial configurations of point vortices. We also find that the enstrophy always dissipates in the sense of distributions at the collapse time. This indicates that the triple collapse is a mechanism for the anomalous enstrophy dissipation in non-smooth inviscid and incompressible flows. Furthermore, it is an interesting example elucidating the emergence of the irreversibility of time in a Hamiltonian dynamical system.
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References
Aref, H.: Motion of three vortices. Phys. Fluids 22(3), 393–400 (1979)
Batchelor, G.K.: Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids Suppl. II(12), 233–239 (1969)
Delort, J.-M.: Existence de nappe de tourbillion en dimension deux. J. Am. Math. Soc. 4, 553–586 (1991)
Diperna, R.J., Majda, A.J.: Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Math. 40, 301–345 (1987)
Eyink, G.L.: Dissipation in turbulent solutions of 2D Euler equations. Nonlinearity 14, 787–802 (2001)
Gotoda, T., Sakajo, T.: Enstrophy variations in the incompressible 2D Euler flows and \(\alpha \) point vortex system. In: Proceedings of the International Conference on Mathematics Fluid Dynamics (2015)
Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler–Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett. 80, 4173–4177 (1998)
Holm, D.D., Marsden, J.E., Ratiu, T.S.: Euler–Poincaré equations and semidirect products with applications to continuum theories. Adv. Math. 137, 1–81 (1998)
Kimura, Y.: Similarity solution of two-dimensional point vortices. J. Phys. Soc. Jpn. 56, 2024–2030 (1987)
Kraichnan, R.H.: Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423 (1967)
Kudela, H.: Collapse of \(n\)-point vortices in self-similar motion. Fluid Dyn. Res. 46, 031414 (2014). doi:10.1088/0169-5983/46/3/031414
Leith, C.E.: Diffusion approximation for two-dimensional turbulence. Phys. Fluids 11, 671–673 (1968)
Lunasin, E., Kurien, S., Taylor, M.A., Titi, E.S.: A study of the Navier–Stokes-\(\alpha \) model for two-dimensional turbulence. J. Turbul. 8, 1–21 (2007)
Majda, A.J.: Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42, 921–939 (1993)
Marchioro, C., Pulvirenti, M.: Mathematical Theory of Incompressible Nonviscous Fluids, Applied Mathematical Sciences, vol. 96. Springer, New York (1994)
Marsden, J.E., Shkoller, S.: The anisotropic Lagrangian averaged Euler and Navier–Stokes equations. Arch. Ration. Mech. Anal. 166, 27–46 (2003)
Newton, P.K.: The N-vortex Problem, Analytical Techniques. Springer, New York (2001)
Novikov, E.A.: Dynamics and statistic of a system of vortices. Sov. Phys. JETP. 41, 937–943 (1976)
Oliver, M., Shkoller, S.: The vortex blob method as a second-grade non-Newtonian fluid. Commun. Part. Differ. Equ. 26, 295–314 (2001)
Sakajo, T.: Instantaneous energy and enstrophy variations in Euler-alpha point vortices via triple collapse. J. Fluid Mech. 702, 188–214 (2012)
Shkoller, S.: Geometry and curvature of diffeomorphism groups with \(H^1\) metric and mean hydrodynamics. J. Funct. Anal. 160, 337–365 (1998)
Watson, G.N.: A Treatise on the Theory of Bessel functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1995)
Yudovich, V.I.: Nonstationary motion of an ideal incompressible liquid. USSR Comput. Math. Phys. 3, 1407–1456 (1963)
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This work was partially supported by JSPS KAKENHI (Grant Number 26287023).
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Communicated by Paul Newton.
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Gotoda, T., Sakajo, T. Distributional Enstrophy Dissipation Via the Collapse of Three Point Vortices. J Nonlinear Sci 26, 1525–1570 (2016). https://doi.org/10.1007/s00332-016-9312-y
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DOI: https://doi.org/10.1007/s00332-016-9312-y