Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T01:26:51.515Z Has data issue: false hasContentIssue false
  • English
  • Français

Momentum transport by gravity waves in a perfectly conducting shear flow

Published online by Cambridge University Press:  29 March 2006

N. Rudraiah  and
M. Venkatachalappa
N. Rudraiah
Affiliation:
Department of Mathematics (Post-Graduate Studies), University Visvesvaraya College of Engineering, Bangalore University, Bangalore
M. Venkatachalappa
Affiliation:
Department of Mathematics (Post-Graduate Studies), University Visvesvaraya College of Engineering, Bangalore University, Bangalore
Get access Rights & Permissions [Opens in a new window]

Abstract

Alfvén-gravitational waves propagating in a Boussinesq, inviscid, adiabatic, perfectly conducting fluid in the presence of a uniform aligned magnetic field in which the mean horizontal velocity U(z) depends on height z only are considered. The governing wave equation has three singularities, at the Doppler-shifted frequencies Ωd = 0, ± ΩA, where ΩA is the Alfvén frequency. Hence the effect of the Lorentz force is to introduce two more critical levels, called hydromagnetic critical levels, in addition to the hydrodynamic critical level. To study the influence of magnetic field on the attenuation of waves two situations, one concerning waves far away from the critical levels (i.e. Ωd [Gt ] ΩA) and the other waves at moderate distances from the critical levels (i.e. Ωd > ΩA), are investigated. In the former case, if the hydrodynamic Richardson number JH exceeds one quarter the waves are attenuated by a factor exp{−2π(JH −¼)½} as they pass through the hydromagnetic critical levels, at which Ωd = ± ΩA, and momentum is transferred to the mean flow there. Whereas in the case of waves at moderate distances from the critical levels the ratio of momentum fluxes on either side of the hydromagnetic critical levels differ by a factor exp {−2π(J −¼)½}, where J (> ¼) is the algebraic sum of hydrodynamic and hydromagnetic Richardson numbers. Thus the solutions to the hydromagnetic system approach asymptotically those of the hydrodynamic system sufficiently far on either side of the magnetic critical layers, though their behaviour in the vicinity of such levels is quite dissimilar. There is no attenuation and momentum transfer to the mean flow across the hydrodynamic critical level, at which Ωd = 0. The general theory is applied to a particular problem of flow over a sinusoidal corrugation. This is significant in considering the propagation of Alfvén-gravity waves, in the presence of a geomagnetic field, from troposphere to ionosphere.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 54 , Issue 2 , 25 July 1972 , pp. 217 - 240
Copyright
© 1972 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

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Tables. Dover.
Booker, J. R. & Bretherton, F. P. 1967 J. Fluid Mech. 27, 513.
Bretherton, F. P. 1966 Quart. J. Roy. Met. Soc. 92, 466.
Bretherton, F. P. 1969 Quart. J. Roy. Met. Soc. 95, 213.
Bretherton, F. P., Hazel, P., Thorpe, S. A. & Wood, I. R. 1967 Appendix to ‘The effect of viscosity and heat conduction of internal gravity waves at a critical level’, by P, Hazel. J. Fluid Mech. 30, 781.Google Scholar
Goldstein, S. 1931 Proc. Roy Soc. A, 132, 524.
Hazel, P. 1967 J. Fluid Mech. 30, 775.
Jeffreys, H. & Jeffreys, B. S. 1946 Methods of Mathematical Physics. Cambridge University Press.
Lighthill, M. J. 1960 Phil. Trans. Roy. Soc. A, 252, 397.
Jones, W. L. 1967 J. Fluid Mech. 30, 439.
Miles, J. W. 1961 J. Fluid Mech. 10, 496.
Narayan, C. L. 1972 Ph.D. thesis, Bangalore University.
Rudraiah, N. 1970 Astrophys. Soc. Japan, 1, 41.
Rudraiah, N. & Venkatachalappa, M. 1972a J. Fluid Mech. 52, 193.
Rudraiah, N. & Venkatachalappa, M. 1972b J. Fluid Mech. 54, 209.
Synge, J. L. 1933 Trans. Roy. Soc. Can. 27, 1.
Taylor, G. I. 1931 Proc. Roy. Soc. A, 132, 499.