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On the stability of vertical double-diffusive interfaces. Part 1. A single plane interface

Published online by Cambridge University Press:  26 April 2006

I. A. Eltayeb  and
D. E. Loper
I. A. Eltayeb
Affiliation:
Department of Mathematics and Computing, Sultan Qaboos University, Muscat, Sultanate of Oman
D. E. Loper
Affiliation:
Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
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Abstract

This is the first part of a three-part study of the stability of vertically oriented double-diffusive interfaces having an imposed vertical stable temperature gradient. Flow is forced by a prescribed jump of composition across the interfaces. Compositional diffusivity is ignored, while thermal diffusivity and viscosity are finite. In this first part, basic-state solutions are presented and discussed for three configurations: a single plane interface, two parallel interfaces and a circular cylindrical interface.

The stability of a single plane interface is then analysed. It is shown that the presence of the compositional jump gives rise to a new type of three-dimensional instability which occurs for any non-zero forcing. This is in contrast to the thermally driven flow adjacent to a rigid wall, which is unstable only for a finite value of the forcing and results in the growth of a two-dimensional perturbation. The timescale for growth of the new instability is given by \[ \tau = \left[\frac{\nu \rho}{(\Delta \rho)_c}\right]^2\frac{\alpha}{\kappa g}\frac{{\rm d}T}{{\rm d}z}, \] where (Δρ)C is the prescribed jump in composition and dT/dz is the imposed temperature gradient.

The influence of thermal diffusion is to enhance instability, while viscosity is stabilizing for nearly all wavenumbers. The interface is unstable for all finite wavenumbers if the Prandtl number is less than 1.472, while regions of stability in the wavenumber plane develop for small horizontal wavenumber and moderate vertical wavenumber for larger values of the Prandtl number. The neutral stability curves are investigated and the maximum growth rate of instability is identified for the whole range of values of the Prandtl number and its properties are elucidated by comparison with previous studies of flows near heated vertical walls.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 228 , July 1991 , pp. 149 - 181
Copyright
© 1991 Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.
Chapman, D. C. 1986 A simple model of the formation and maintenance of the shelf-slope front in the middle Atlantic bight. J. Phys. Oceanogr. 16, 1273–1279.Google Scholar
Chen, C. F. & Johnson, D. H. 1984 Double-diffusive convection: a report on an engineering foundation conference. J. Fluid Mech. 138, 405–416.Google Scholar
Copley, S. M., Giami, A. F., Johnson, S. M. & Hornbecker, M. F. 1970 On the origin of freckles in unidirectionally solidified castings. Metall. Trans. 1, 2193–2204.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Instability. Cambridge University Press.
Dudis, J. J. & Davis, S. H. 1971 Energy stability of the buoyancy boundary layer. J. Fluid Mech. 47, 381–403.Google Scholar
Eltayeb, I. A. 1972 Hydromagnetic convection in a rapidly rotating layer.. Proc. R. Soc. Lond. A 326, 229–254.Google Scholar
Fowler, A. C. 1985 The formation of freckles in binary alloys. IMA J. Appl. Maths 35, 159–174.Google Scholar
Gill, A. E. 1966 The boundary-layer regime for convection in a rectangular cavity. J. Fluid Mech. 26, 515–536.Google Scholar
Gill, A. E. & Davey, A. 1969 Instabilities of a buoyancy-driven system. J. Fluid Mech. 35, 775–798.Google Scholar
Holyer, J. Y. 1983 Double diffusive interleaving due to horizontal gradients. J. Fluid Mech. 137, 347–362.Google Scholar
Howard, L. N. & Veronis, G. 1987 The salt-finger zone. J. Fluid Mech. 183, 1–23 (referred to herein as H&V).Google Scholar
Hsueh, Y. & Cushman-Roisin, B. 1983 On the formation of surface to bottom fronts over steep topography. J. Geophys. Res. 88, 743–750.Google Scholar
Huppert, H. E. & Moore, D. R. 1976 Nonlinear double diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Loper, D. E. 1989 Dynamo energetics and the structure of the outer core. Geophys. Astrophys. Fluid Dyn. 49, 213–219.Google Scholar
Ou, H. W. 1983 Some two-layer models of the shelf-slope front: geostrophic adjustment and its maintenance. J. Phys. Oceanogr. 13, 1798–1808.Google Scholar
Roberts, P. H. & Loper, D. E. 1983 Towards a theory of the structure and evolution of a dendrite layer. In Stellar and Planetary Magnetism (ed. A. M. Soward), pp. 329–349, Gordon and Breach.
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.
Turner, J. S. 1985 Multicomponent convection. Ann. Rev. Fluid Mech. 17, 11–44.Google Scholar
Wang, D.-P. 1984 Mutual intrusion of a gravity current and density front formation. J. Phys. Oceanogr. 14, 1191–1199.Google Scholar
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