Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-06T01:08:19.290Z Has data issue: false hasContentIssue false
  • English
  • Français

Propagation of hydormagnetic planetary waves on a beta-plane through magnetic and velocity shear

Published online by Cambridge University Press:  12 April 2006

I. A. Eltayeb  and
J. F. Mckenzie
I. A. Eltayeb
Affiliation:
Department of Mathematics, Faculty of Science, University of Khartoum, Sudan
J. F. Mckenzie
Affiliation:
Department of Mathematics, Faculty of Science, University of Khartoum, Sudan
Get access Rights & Permissions [Opens in a new window]

Abstract

The propagation of hydromagnetic planetary waves in a rotating thin shell of fluid in regions of magnetic and velocity shears is studied using the β-plane approximation. In slowly varying shear the use of the WKBJ approximation makes it possible to construct the various types of ray trajectory that can occur and consequently the conditions that give rise to critical-latitude phenomena and trapping are deduced.

The opposite extreme to the WKBJ limit, namely reflexion and refraction of waves by a current-vortex sheet, is also analysed. In this case the conditions that lead to wave amplification (or over-reflexion) are investigated. Qualitatively, it is found that reflected Alfvén modes are amplified if the jump in the flow speed across the sheet lies between two speeds which are respectively greater and less than the sum of the Alfvén speeds on either side of the sheet. Also, Rossby waves incident upon a sufficiently strong easterly flow can suffer over-reflexion.

The general case of reflexion and refraction at a finite double (magnetic and velocity) shear layer is discussed. In analogy with the invariance of ‘wave action’ of gravity waves in a shear flow we construct a quantity [Ascr ] which is invariant except at critical latitudes, where it is discontinuous. By using the asymptotic solutions near these critical latitudes and by adopting the proper matching procedure for the solutions on either side of these latitudes it is possible to relate the two constant values of [Ascr ] on either side of each critical latitude. These general results are then applied to various profiles of shear flow and magnetic field so as to elucidate the manner in which an incident wave is reflected from and transmitted through a double layer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 81 , Issue 1 , 9 June 1977 , pp. 1 - 23
Copyright
© 1977 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Acheson, D. J. 1973 J. Fluid Mech. 58, 27.
Booker, J. R. & Bretherton, F. P. 1967 J. Fluid Mech. 27, 513.
Bretheeton, F. P. & Garrett, C. J. R. 1968 Proc. Roy. Soc. A 302, 529.
Busse, F. H. 1975 Geophys. J. Roy. Astr. Soc. 42, 437.
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford: Clarendon Press.
Eliassen, A. & Palm, E. 1960 Geofys. Publ. 22, 1.
Eltayeb, I. A. 1977 Phil. Trans. 285, 607–636.
Eltayeb, I. A. & McKenzie, J. F. 1975 J. Fluid Mech. 72, 661.
Eskinazi, S. 1967 Vector Mechanics of Fluids and Magnetofluids. Academic.
Grimshaw, R. 1975 J. Fluid Mech. 70, 287.
Hide, R. 1966 Phil. Trans. A 259, 615.
Hide, R. & Jones, M. V. 1972 Met. Office Sci. Paper no. 33. H.M.S.O.
Lighthill, M. J. 1967 I.A.U. Symp. no. 28, p. 429.
Longuet-Higgins, M. S. 1965 Proc. Roy. Soc. A 284, 40.
Mckenzie, J. F. 1970 Planet. Space Sci. 18, 1.
Mckenzie, J. F. 1972 J. Geophys. Res. 77, 2915.
Mckenzie, J. F. 1973 J. Fluid Mech. 58, 709.
Mekki, O. M. & McKenzie, J. F. 1977 To be published.
Miles, J. W. 1961 J. Fluid Mech. 10, 4–96.
Roberts, P. H. & Soward, A. M. 1972 Ann. Rev. Fluid Mech. 4, 117.
Stewartson, K. 1967 Proc. Roy. Soc. A 299, 173.