Abstract
The question whether finite time singularities develop in incompressible hydro- and magnetohydrodynamic systems starting from smooth initial conditions is still an open problem. Here we present numerical simulations using the technique of adaptive mesh refinement which show evidence that in the 3D incompressible Euler equations a finite time blow-up in the vorticity occurs whereas in the 2D incompressible magnetohydrodynamic equations only exponential growth of vorticity and current density is observed.
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References
J. T. Beale, T. Kato, and A. Majda, Remarks on the breakdown of smooth solutions for the 3d euler equations, Comm. Math. Phys. 94 (1984), 61–64.
J. Bell, M. Berger, J. Saltzman, and M. Welcome, Three-dimensional adaptive mesh refinement for hyperbolic conservation laws, SIAM J. Sci. Comput. 15(1) (1994), 127–138.
J. B. Bell, P. Colella, and H. M. Glaz, A second-order projection method for the incompressible navier-stokes equation, J. Comput. Phys. 85 (1989), 257–283.
J. B. Bell and D. L. Marcus, Vorticity intensification and transition to turbulence in the three-dimensional euler equation, Comm. Math. Phys. 147 (1992), 371–394.
M. J. Berger and P. Colella, Local adaptive mesh refinement for shock hydodynamics, J. Comput. Phys. 82 (1989), 64–84.
D. Biskamp and H. Welter, Dynamics of decaying two-dimensional magnetohydrodynamic turbulence, Phys. Fluids B 1 (1989), 1964–1979.
O. N. Boratav and R. B. Pelz, Locally isotropic pressure hessian in a high-symmetry flow, Phys. Fluids 7(5) (1995), 895–897.
M. E. Brächet, M. Meneguzzi, A. Vincent, H. Politano, and P. L. Sulem, Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows, Phys. Fluids A 4 (1992), 2845–2854.
H. Friedel, R. Grauer, and C. Marliani, Adaptive mesh refinement for singular current sheets in incompressible magnetohydrodynamic flows, J. Comput. Phys. 134 (1997), 190–198.
R. Grauer and C. Marliani, Numerical and analytical estimates for the structure functions in two-dimensional magnetohydrodynamic flows, Phys. Plasmas 2(1) (1995), 41–47.
R. Grauer, C. Marliani, and K. Germaschewski, Adaptive mesh refinement for singular solutions of the incompressible euler equations, submitted to Phys. Rev. Lett. (1997).
W. Hackbusch, Iterative solution of large sparse systems of equations, Applied mathematical sciences, Springer, New York, 1994.
R. M. Kerr, Evidence for a singularity of the three-dimensional, incompressible euler equations, Phys. Fluids A 5(7) (1993), 1725–1746.
A. Pouquet, Turbulence, statistics and structures: An introduction, Plasma Astrophysics (Berlin, Heidelberg) (C. Chiuderi and G. Einaudi, eds.), Lecture Notes in Physics, 468, Springer-Verlag, 1996, 163–212.
P. L. Sulem, U. Frisch, A. Pouquet, and M. Meneguzzi, On the exponential flattening of current sheets near neutral x-points in two-dimensional ideal mhd flow, J. Plasma Phys. 33 (1985), 191–198.
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© 1999 Springer Basel AG
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Grauer, R. (1999). Adaptive Mesh Refinement for Singular Structures in Incompressible Hydro- and Magnetohydrodynamic Flows. In: Fey, M., Jeltsch, R. (eds) Hyperbolic Problems: Theory, Numerics, Applications. International Series of Numerical Mathematics, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8720-5_44
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DOI: https://doi.org/10.1007/978-3-0348-8720-5_44
Publisher Name: Birkhäuser, Basel
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