Vortex Line Representation for the Hydrodynamic Type Equations

References

• Arnold , V I . 1989 . Mathematical Methods in Classical Mechanics , 2nd , Berlin : Springer .
   Google Scholar
• Arnold , V I . 1981 . Theory of Catastrophe , Moscow : Znanie . (in Russian) [English transl.: Theory of Catastrophe, Springer, 1986]
   Google Scholar
• Boratav , O N and Pelz , R B . 1994 . Direct numerical simulation of transition to turbulence from high-symmetry initial condition . Phys. Fluids , 6 : 2757
   Web of Science ®Google Scholar
• Brachet , M E , Meneguzzi , M , Vincent , A , Politano , H and Sulem , P L . 1992 . Numerical evidence of smooth self-similar dynamics and possibility of subsequent collapse for three-dimensional ideal flows . Phys. Fluids A , 4 : 2845
   Web of Science ®Google Scholar
• Da Rios , L S . 1906 . Rend. Circ. Mat. Palerno , 22 : 117 Betchov R., J. Fluid Mech. 22 (1965) 471
   Google Scholar
• Grauer , R , Marliani , C and Germaschewski , K . 1998 . Adaptive mesh refinement for singular solutions of the incompressible Euler equations . Phys. Rev. Lett. , 80 : 4177
   Web of Science ®Google Scholar
• Hasimoto , R. 1972 . A soliton on vortex filament . J. Fluid Mech. , 51 : 477
   Web of Science ®Google Scholar
• Kerr , R M . 1993 . Evidence for a singularity of the 3-dimensional, incompressible Euler equations . Phys. Fluids A , 5 : 1725
   Web of Science ®Google Scholar
• Kerr , R M . 1999 . The outer regions in singular Euler, in Fundamental problematic issues in turbulence , Edited by: Tsinober , A and Gyr , R . Vol. 41 , Basel : Birkhauser .
   Google Scholar
• Kuznetsov , E A and Mikhailov , A V . 1980 . On the topological meaning of canonical Clebsch variables . Phys. Lett. A , 77 : 37
   Web of Science ®Google Scholar
• Kuznetsov , E A and Ruban , V P . 1998 . Hamiltonian dynamics of vortex lines for systems of the hydrodynamic type . Pis’ma v ZhETF , 67 : 1015 [JETP Letters 67 (1998), 1076]
   Web of Science ®Google Scholar
• Kuznetsov , E A and Ruban , V P . 2000 . Hamiltonian dynamics of vortex and magnetic lines in the hydrodynamic type models . Phys. Rev. E , 61 : 831 – 841 .
   Web of Science ®Google Scholar
• Kuznetsov , E A . 2002 . Vortex line representation for flows of ideal and viscous fluids . Pis’ma v ZHETF , 76 : 406 – 410 . [JETP Letters 76 (2002), 346–350]
   Web of Science ®Google Scholar
• Kuznetsov , E A , Podvigina , O N and Zheligovsky , V A . 2003 . Numerical evidence of breaking of vortex lines in an ideal fluid . Fluid dynamics and its application , 71 : 305 – 316 . Tubes, Sheets and Singularities in Fluid Dynamics. eds. K. Bajer, H.K. Moffatt, Kluwer
   Google Scholar
• Kuznetsov , E A and Ruban , V P . 2000 . Collapse of vortex lines in hydrodynamics . ZhETF , 118 : 893 [JETP 91 (2000), 776]
   Google Scholar
• Kuznetsov , E A , Passot , T and Sulem , P L . 2004 . Compressible dynamics of magnetic field lines for incompressible MHD flows . Physics of Plasmas , 11 : 1410
   Web of Science ®Google Scholar
• Kuznetsov , E A , Naulin , V , Nielsen , A H and Rasmussen , J J . Sharp vorticity gradients in two-dimensional hydrodynamic turbulence . arXiv:physics/0407101
   Google Scholar
• Lamb , H . Hydrodynamics, , 6th , New York : Dover .
   Google Scholar
• Moreau , J J . 1961 . C. R. Acad. Sc. Paris , 252 : 2810 Moffatt H K, The degree of knottedness of tangled vortex lines, J. Fluid Mech. 35 (1969), 117.
   Google Scholar
• Pelz , R B . 1997 . Locally self-similar, finite-time collapse in a high-symmetry vortex filament model . Phys. Rev. E , 55 : 1617
   Web of Science ®Google Scholar
• Ricca , R L . 1991 . Redicovery of Da Rios equations . Nature , 352 : 561
   Web of Science ®Google Scholar
• Ruban , V P . 1999 . Motion of magnetic flux lines in magnetohydrodynamics . ZhETF , 116 : 563 [JETP 89 (1999), 299]
   Google Scholar
• Ruban , V P . 2002 . Current-sheet formation in incompressible electron magnetohydrodynamics . Phys. Rev. E , 65 047401
   Web of Science ®Google Scholar
• Salmon , R . 1988 . Hamiltonian Fluid Mechanics . Ann. Rev. Fluid Mech , 20 : 225
   Web of Science ®Google Scholar
• Shandarin , S F and Zeldovich , Ya B . 1989 . The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating medium . Rev. Mod. Phys. , 61 : 185
   Web of Science ®Google Scholar
• Zakharov , V E and Kuznetsov , E A . 1997 . Hamiltonian formalism for nonlinear waves . UFN , 167 : 1137 – 1168 . [Physics-Uspekhi 40 (1997), 1087–1116]
   Google Scholar
• Zakharov , V E and Takhtajan , L A . 1979 . Equivalence of a nonlinear Schrodinger equation and Heisenberg ferromagnetic equation . Teor Mat. Fiz , 38 : 26 (Theor. Math. Phys.)
   Google Scholar
• Zheligovsky , V A , Kuznetsov , E A and Podvigina , O M . 2001 . Numerical modeling of collapse in ideal incompressible hydrodynamics . Pis’ma v ZhETF , 74 : 402 [JETP Letters 74 (2001), 367]
   Web of Science ®Google Scholar