## The geodesic flow generates a fast dynamo: an elementary proof

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- by C. Chicone and Y. Latushkin PDF
- Proc. Amer. Math. Soc.
**125**(1997), 3391-3396 Request permission

## Abstract:

We give elementary and explicit arguments to show that the geodesic flow on the unit tangent bundle of a two dimensional Riemannian manifold with constant negative curvature provides an example of a “fast” dynamo for the magnetic kinematic dynamo equation.## References

- Ralph Abraham, Jerrold E. Marsden, and Tudor S. Raţiu,
*Manifolds, tensor analysis, and applications*, Global Analysis Pure and Applied: Series B, vol. 2, Addison-Wesley Publishing Co., Reading, Mass., 1983. MR**697563** - V. Arnold,
*Les méthodes mathématiques de la mécanique classique*, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR**0474391** - V. I. Arnold, Ya. B. Zel$^\prime$dovich, A. A. Rasumaikin, and D. D. Sokolov,
*Magnetic field in a stationary flow with stretching in Riemannian space,*Sov. Phys. JETP,**54**(6) (1981) 1083–1086. - B. Bayly,
*Fast magnetic dynamos in chaotic flows,*Phys. Rev. Lett.**57**(22) (1986) 2800. - B. J. Bayly and S. Childress,
*Fast-dynamo action in unsteady flows and maps in three dimensions*, Phys. Rev. Lett.**59**(1987), no. 14, 1573–1756. MR**908623**, DOI 10.1103/PhysRevLett.59.1573 - J. Beem, C. Chicone, and P. Ehrlich,
*The geodesic flow and sectional curvature of pseudo-Riemannian manifolds*, Geom. Dedicata**12**(1982), no. 2, 111–118. MR**648686**, DOI 10.1007/BF00147630 - Carmen Chicone,
*The topology of stationary curl parallel solutions of Euler’s equations*, Israel J. Math.**39**(1981), no. 1-2, 161–166. MR**617299**, DOI 10.1007/BF02762862 - Carmen C. Chicone,
*Tangent bundle connections and the geodesic flow*, Rocky Mountain J. Math.**11**(1981), no. 2, 305–317. MR**619678**, DOI 10.1216/RMJ-1981-11-2-305 - Carmen Chicone and Paul Ehrlich,
*Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds*, Manuscripta Math.**31**(1980), no. 1-3, 297–316. MR**576502**, DOI 10.1007/BF01303279 - C. Chicone, Y. Latushkin, and S. Montgomery-Smith,
*The spectrum of the kinematic dynamo operator for an ideally conducting fluid*, Comm. Math. Phys.**173**(1995), no. 2, 379–400. MR**1355630** - J.M. Finn, and E. Ott,
*Chaotic flows and magnetic dynamos,*Phys. Rev. Lett.**60**(9) (1988) 760–763. - L. Green,
*Geodesic Flows,*Lecture Notes in Math.,**200**(1971) 25–27. - Leon W. Green,
*When is an Anosov flow geodesic?*, Ergodic Theory Dynam. Systems**12**(1992), no. 2, 227–232. MR**1176620**, DOI 10.1017/S0143385700006714 - A.M. Soward,
*Fast dynamo actions in a steady flow,*Journ. Fluid Mech.**180**(1987) 267–295. - M. M. Vishik,
*Magnetic field generation by the motion of a highly conducting fluid*, Geophys. Astrophys. Fluid Dynam.**48**(1989), no. 1-3, 151–167. MR**1024693**, DOI 10.1080/03091928908219531 - M. M. Vishik,
*A system of equations arising in magnetohydrodynamics*, Dokl. Akad. Nauk SSSR**275**(1984), no. 6, 1295–1299 (Russian). MR**746372**

## Additional Information

**C. Chicone**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- Email: carmen@chicone.math.missouri.edu
**Y. Latushkin**- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 213557
- Email: yuri@math.missouri.edu
- Received by editor(s): April 24, 1996
- Additional Notes: The first author’s research was supported by the National Science Foundation under the grant DMS-9303767; the second author was supported by the National Science Foundation under the grant DMS-9400518 and by the SRF of the University of Missouri.
- Communicated by: Jeffrey B. Rauch
- © Copyright 1997 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**125**(1997), 3391-3396 - MSC (1991): Primary 76W05, 58F99, 58G25
- DOI: https://doi.org/10.1090/S0002-9939-97-04187-7
- MathSciNet review: 1443147