Abstract
We consider the existence of steady incompressible fluids (solutions to the Euler equations) on Riemannian manifolds of dimensions three and higher. We demonstrate that, as in the case of the ABC fields in dimension three, there exist chaotic Beltrami fields – nonsingular eigenfields of the curl operator – in higher dimensions. We give an explicit set of analytic examples on a non-Euclidean five-torus T 5. We also detail a ‘plug’ construction for inserting chaotic vortices into a Beltrami field. These constructions employ contact-topological techniques.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[AK98] Arnold, V. I. and Khesin, B.: Topological Methods in Hydrodynamics, Springer-Verlag, Berlin, 1998.
[Arn66] Arnold, V. I.: Sur la géométrie differentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier 16 (1966), 316–361.
[BCG91] Bryant, R. L., Chern, S. S., Gardner, R. B., Goldschmidt, H. L. and Griffiths, P. A.: Exterior Differential Systems, Springer-Verlag, New York, 1991.
[DFG86] Dombre, T., Frisch, U., Greene, J., Hénon, M., Mehr, A. and Soward, A.: Chaotic streamlines in the ABC flows, J. Fluid Mech. 167 (1986), 353–391.
[EG00a] Etnyre, J. and Ghrist, R.: Contact topology and hydrodynamics I: Beltrami fields and the Seifert conjecture, Nonlinearity 13 (2000), 441–458.
[EG00b] Etnyre, J. and Ghrist, R.: Contact topology and hydrodynamics III: knotted flowlines, Trans. Amer. Math. Soc. 352 (2000), 5781–5794.
[EG99] Etnyre, J. and Ghrist, R.: Stratified integrals and unknots in inviscid flows, Contempt. Math. 246 (1999), 99–112.
[Gau85] Gautero, J.-L.: Chaos lagrangien pour une classe d'écoulements de Beltrami, C.R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 301(15) (1985), 1095–1098.
[GK94] Ginzburg, V. and Khesin, B.: Steady fluid flows and symplectic geometry, J. Geom. Phys. 14 (1994), 195–210.
[Güm97] Gümral, H.: Lagrangian description, symplectic structure, and invariants of 3D fluid flow, Phys. Lett. A 232(6) (1997), 417–424.
[HZD98] Huang, D.-B., Zhao, X.-H. and Dai, H.-H.: Invariant tori and chaotic streamlines in the ABC flow, Phys. Lett. A 237(3) (1998), 136–140.
[Lut77] Lutz, R.: Structures de contact sur les fibrés principaux en cercles de dimension trois, Ann. Inst. Fourier, 3 (1977), 1–15.
[Mar71] Martinet, J.: Formes de contact sur les variétés de dimension 3, Lecture Notes in Math. 299, Springer, New York, 1971, pp. 142–163.
[MS95] McDuff, D. and Salamon, D.: Introduction to Symplectic Topology, Oxford Univ. Press, New York, 1995.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Ghrist, R. Steady Nonintegrable High-Dimensional Fluids. Letters in Mathematical Physics 55, 193–204 (2001). https://doi.org/10.1023/A:1010936025007
Issue Date:
DOI: https://doi.org/10.1023/A:1010936025007