A quaternionic structure in the three-dimensional Euler and ideal magneto-hydrodynamics equations

Physica D: Nonlinear Phenomena

Volumn 166, Issue 1-2, 2002, Pages 17-28

  Gibbon, J D ^a  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Euler; Ideal MHD; Quaternions; Riccati; Zero eigenvalue Schr dinger

Indexed keywords

EID: 0036609953     PISSN: 01672789     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0167-2789(02)00434-7     Document Type: Article

References (31)

1. V.N. Roubtsov I. Roulstone Examples of quaternionic and Kähler structures in Hamiltonian models of nearly geostrophic flow J. Phys. A 30 1997 L63 L68
2. V.N. Roubtsov, I. Roulstone, Holomorphic structures in hydrodynamical models of nearly geostrophic flow, Proc. Roy. Soc. London 457 (2001) 1519–1531.
3. B. Galanti J.D. Gibbon M. Heritage Vorticity alignment results for the 3D Euler and Navier–Stokes equations Nonlinearity 10 1997 1675 1695
4. J.D. Gibbon, B. Galanti, R. Kerr, Stretching and compression of vorticity in the 3D Euler, in: J.C.R. Hunt, J.C. Vassilicos (Eds.), Turbulence Structure and Vortex Dynamics, Cambridge University Press, Cambridge, 2000, pp. 23–34.
5. H. Ertel Ein Neuer Hydrodynamischer Wirbelsatz Met. Z. 59 1942 271 281
6. B.J. Hoskins M.E. McIntyre A.W. Robertson On the use and significance of isentropic potential vorticity maps Quart. J. Roy. Met. Soc. 111 1985 877 946
7. A. Viudez, On the relation between Beltrami’s material vorticity and Rossby–Ertel’s potential, J. Atmos. Sci. (2001), in press.
8. A. Viudez On Ertel’s potential vorticity theorem, On the impermeability theorem for potential vorticity J. Atmos. Sci. 56 1999 507 516
9. C.-G. Rossby, Planetary flow patterns in the atmosphere, Quart. J. Roy. Met. Soc. 66 (2) 68–87.
10. E. Beltrami Sui principii fondamentali dell’ idrodinamica razionali Memorie della Accademia delle Scienze dell’Istituto di Bologna 1 1871 431 476
11. C. Truesdell, R.A. Toupin, in: S. Flugge (Ed.), Classical Field Theories, Encyclopaedia of Physics, Vol. III, No. 1, Springer, Berlin, 1960.
12. K. Ohkitani S. Kishiba Nonlocal nature of vortex stretching in an inviscid fluid Phys. Fluids A 7 1995 411
13. K. Ohkitani Eigenvalue problems in three-dimensional Euler flows Phys. Fluids A 5 1993 2570
14. M. Adler J. Moser On a class of polynomials connected with the Korteweg de Vries equation Commun. Math. Phys. 61 1978 1 30
15. P. Constantin Geometric statistics in turbulence SIAM Rev. 36 1994 73
16. A. Vincent M. Meneguzzi The dynamics of vorticity tubes of homogeneous turbulence J. Fluid Mech. 225 1994 245 254
17. R. Kerr Evidence for a singularity of the 3-dimensional, incompressible Euler equations Phys. Fluids A 5 1993 1725
18. U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
19. P. Vieillefosse Internal motion of a small element of fluid in an inviscid flow Physica A 125 1984 837
20. P. Constantin Ch. Fefferman A. Majda Geometric constraints on potentially singular solutions for the 3D Euler equations Commun. Partial Diff. Eqs. 21 1996 559 571
21. W. Ashurst W. Kerstein R. Kerr C. Gibson Alignment of vorticity and scalar gradient with strain rate in simulated Navier–Stokes turbulence Phys. Fluids 30 1987 2343
22. J. Jiminez Kinematic alignments in turbulent flows Phys. Fluids A 4 1992 652 654
23. A. Tsinober E. Kit T. Dracos Experimental investigation of the field of velocity gradients in turbulent flows J. Fluid Mech. 242 1992 169
24. A. Tsinober, L. Shtilman, A. Sinyavskii, H. Vaisburd, Vortex stretching and enstrophy generation in numerical and laboratory turbulence, in: M. Meneguzzi, A. Pouquet, P.-L. Sulem (Eds.), Small Scale Structures in Three-dimensional Hydro- and MHD Turbulence, Vol. 462, Lecture Notes in Physics, Springer, Berlin, 1995, p. 17.
25. H.K. Moffatt Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology J. Fluid Mech. 159 1985 359 378
26. H.K. Moffatt Energy spectrum of a knotted magnetic flux tube Nature 347 1990 367 369
27. J. Leray Essai sur le mouvement d’un liquide visquex emplissant l’espace Acta Math. 63 1934 193 248
28. R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Applied Mathematical Sciences, Vol. 68, Springer, New York, 1988.
29. P. Constantin, C. Foias, Navier–Stokes Equations, University of Chicago Press, Chicago, 1988.
30. L. Caffarelli R. Kohn L. Nirenberg Commun. Pure Appl. Math. 35 1982 771
31. C.R. Doering, J.D. Gibbon, Applied analysis of the Navier–Stokes equations, Cambridge University Press, Cambridge, 1995.