ABC flows then and now

References

• Alexakis , A . 2011 . Searching for the fastest dynamo: laminar ABC flows . Phys. Rev. E , 84 : 026321-1 – 02631-10 .
   Google Scholar
• Archontis , V . Linear, non-linear and turbulent dynamos. Ph.D. Thesis, University of Copenhagen, 2000. Available online at: www.astro.ku.dk/~bill
   Google Scholar
• Archontis , V , Dorch , B and Nordlund , A . 2003 . Numerical simulations of kinematic dynamo action . Astron. Astrophys. , 397 : 393 – 401 .
   Web of Science ®Google Scholar
• Archontis , V , Dorch , B and Nordlund , A . 2007 . Nonlinear MHD dynamo operating at equipartition . Astron. Astrophys. , 472 : 715 – 726 .
   Web of Science ®Google Scholar
• Arnold , VI . 1965 . Sur la topologie des écoulements stationnaires des fluides parfaits . C. R. Acad. Sci. Paris , 261 : 17 – 20 .
   Google Scholar
• Arnold , VI and Korkina , EI . 1983 . The growth of a magnetic field in a three-dimensional steady incompressible flow . Vest. Mosk. Un. Ta. Ser 1 Mat. Mecanika , 3 : 43 – 46 . (in Russian)
   Google Scholar
• Ashwin , P and Podvigina , O . 2003 . Hopf bifurcation with cubic symmetry and instability of the ABC flow . Proc. R. Soc. Lond. A , 459 : 1801 – 1827 .
   Web of Science ®Google Scholar
• Bayly , BJ and Childress , S . 1988 . Construction of fast dynamos using unsteady flows and maps in three dimensions . Geophys. Astrophys. Fluid Dyn. , 44 : 211 – 240 .
   Web of Science ®Google Scholar
• Bayly , VI and Childress , S . 1989 . Unsteady dynamo effects at large magnetic Reynolds number . Geophys. Astrophys. Fluid Dyn. , 49 : 23 – 43 .
   Web of Science ®Google Scholar
• Brown , BP , Browning , MK , Brun , AS , Miesch , MS and Toomre , J . 2010 . Persistent magnetic wreaths in a rapidly rotating Sun . Astrophys. J. , 711 : 424 – 438 .
   Web of Science ®Google Scholar
• Brummell , N , Cattaneo , F and Tobias , SM . 2001 . Linear and nonlinear dynamo properties of time-dependent ABC flows . Fluid Dyn. Res. , 28 : 237 – 265 .
   Web of Science ®Google Scholar
• Cameron , R and Galloway , D . 2006a . Saturation properties of the Archontis dynamo . Mon. Not. R. Astron. Soc. , 365 : 735 – 746 .
   Web of Science ®Google Scholar
• Cameron , R and Galloway , D . 2006b . High field strength modified ABC and rotor dynamos . Mon. Not. R. Astron. Soc. , 367 : 1163 – 1169 .
   Web of Science ®Google Scholar
• Childress , S and Gilbert , AD . 1995 . Stretch, Twist, Fold: The Fast Dynamo , Berlin : Springer-Verlag .
   Google Scholar
• Dombre , T , Frisch , U , Greene , JM , Hénon , M , Mehr , A and Soward , AM . 1986 . Chaotic streamlines in the ABC flows . J. Fluid Mech. , 167 : 353 – 391 .
   Web of Science ®Google Scholar
• Dorch , SBF . 2000 . On the structure of the magnetic field in a kinematic ABC dynamo . Phys. Scr. , 61 : 717 – 722 .
   Web of Science ®Google Scholar
• Friedlander , SJ and Vishik , MM . 1995 . Rigorous bounds on the fast dynamo growth rate involving topological entropy . Chaos , 5 : 416 – 423 .
   PubMed Web of Science ®Google Scholar
• Gailitis , A . 1970 . Self-excitation of a magnetic field by a pair of annular vortices . Magnetohydrodynamics , 6 : 14 – 17 .
   Google Scholar
• Galanti , B , Sulem , PL and Pouquet , A . 1992 . Linear and nonlinear dynamos associated with ABC flows . Geophys. Astrophys. Fluid Dyn. , 66 : 183 – 208 .
   Web of Science ®Google Scholar
• Galloway , DJ . 2003 . “ Fast dynamos ” . In Advances in Nonlinear Dynamos , Edited by: Ferriz-Mas , A and Nunez , M . 37 – 59 . London and New York : Taylor & Francis .
   Google Scholar
• Galloway , DJ and Frisch , U . 1986 . Dynamo action in a family of flows with chaotic streamlines . Geophys. Astrophys. Fluid Dyn. , 36 : 53 – 83 .
   Web of Science ®Google Scholar
• Galloway , DJ and Frisch , U . 1987 . A note on the stability of a family of space-periodic Beltrami flows . J. Fluid Mech. , 180 : 557 – 564 .
   Web of Science ®Google Scholar
• Galloway , DJ and O'Brian , NR . 1993 . “ Numerical calculations of dynamos for ABC and related flows ” . In Solar and Planetary Dynamos , Edited by: Proctor , MRE , Matthews , PC and Rucklidge , AM . 105 – 113 . Cambridge : Cambridge University Press .
   Google Scholar
• Galloway , DJ and Proctor , MRE . 1992 . Numerical calculations of fast dynamos in smooth velocity fields with realistic diffusion . Nature , 356 : 691 – 693 .
   Web of Science ®Google Scholar
• Galloway , DJ and Zheligovsky , VA . 1994 . On a class of non-axisymmetric solutions to the electromagnetic induction equation . Geophys. Astrophys. Fluid Dyn. , 76 : 253 – 264 .
   Web of Science ®Google Scholar
• Gibson , RD . 1968 . The herzenberg dynamo II . Q.J. Mech. Appl. Math. , 21 : 257 – 267 .
   Web of Science ®Google Scholar
• Gilbert , AD . 1991 . Fast dynamo action in a steady chaotic flow . Nature , 350 : 483 – 485 .
   Web of Science ®Google Scholar
• Gilbert , AD . 2011 . Ponty, Y. and Zheligovsky, V., Dissipative structures in a nonlinear dynamo . Geophys. Astrophys. Fluid Dyn. , 105 : 629 – 653 .
   Web of Science ®Google Scholar
• Hénon , M . 1966 . Sur la topologie des lignes de courant dans un cas particulier . C. R. Acad. Sci. Paris , 263 : 312 – 314 .
   Google Scholar
• Hollerbach , R , Galloway , DJ and Proctor , MRE . 1995 . Numerical evidence of fast dynamo action in a spherical shell . Phys. Rev. Lett. , 74 : 3145 – 3148 .
   PubMed Web of Science ®Google Scholar
• Hollerbach , R , Galloway , DJ and Proctor , MRE . 1998 . On the adjustment to the Bondi-gold theorem in a spherical shell fast dynamo . Geophys. Astrophys. Fluid Dyn. , 87 : 111 – 132 .
   Web of Science ®Google Scholar
• Klapper , I and Young , LS . 1995 . Rigorous bounds on the fast dynamo growth rate involving topological entropy . Comm. Math. Phys. , 173 : 623 – 646 .
   Web of Science ®Google Scholar
• Krause , F and Rädler , K-H . 1980 . Mean-Field Magnetohydrodynamics and Dynamo Theory , Pergamon : Oxford .
   Google Scholar
• Lau , Y-T and Finn , JM . 1993 . Fast dynamos with finite resistivity in steady flows with stagnation points . Phys. Fluids B , 5 : 365 – 375 .
   Google Scholar
• Mininni , PD . 2007 . Inverse cascades and α effect at a low magnetic Prandtl number . Phys. Rev. E , 76 : 026316 – 1-026316-9 .
   Web of Science ®Google Scholar
• Moffatt , HK and Proctor , MRE . 1985 . Topological constraints associated with fast dynamo action . J. Fluid Mech. , 154 : 493 – 507 .
   Web of Science ®Google Scholar
• Otani , NF . 1993 . A fast kinematic dynamo in two-dimensional time-dependent flows . J. Fluid Mech. , 253 : 327 – 349 .
   Web of Science ®Google Scholar
• Podvigina , OM and Pouquet , A . 1994 . On the non-linear stability of the 1:1:1 ABC flow . Physica D , 75 : 471 – 508 .
   Web of Science ®Google Scholar
• Ponty , Y , Pouquet , A and Sulem , PL . 1995 . Dynamo action in weakly chaotic two-dimensional flows . Geophys. Astrophys. Fluid Dyn. , 79 : 239 – 257 .
   Web of Science ®Google Scholar
• Proctor , MRE . 1977 . On Backus' necessary condition for dynamo action in a conducting sphere . Geophys. Astrophys. Fluid Dyn. , 9 : 89 – 93 .
   Google Scholar
• Roberts , GO . 1970 . Spatially periodic dynamos . Philos. Trans. R. Soc , A266 : 535 – 558 .
   Web of Science ®Google Scholar
• Soward , AM . 1987 . Fast dynamos in a steady flow . J. Fluid Mech. , 180 : 267 – 295 .
   Web of Science ®Google Scholar
• Teyssier , R , Fromang , S and Dormy , E . 2006 . Kinematic dynamos using constrained transport with high order Godunov schemes and adaptive mesh refinement . J. Comput. Phys. , 218 : 44 – 67 .
   Web of Science ®Google Scholar
• Vainshtein , S and Cattaneo , F . 1992 . Nonlinear restrictions on dynamo action . Astrophys. J. , 393 : 479 – 484 .
   Web of Science ®Google Scholar
• Vainshtein , S and Zeldovich , YaB . 1972 . Origin of magnetic fields in astrophysics . Sov. Phys. Usp. , 15 : 159 – 172 .
   Google Scholar