The Influence of Boundaries on the Stability of Compositional Plumes


The influence of boundaries on the dynamics of a compositional plume is studied using a simple model in which a column of buoyant fluid rises in a less buoyant fluid bounded by two vertical walls with a finite distance apart. The problem is governed by four dimensionless parameters: The Grashoff number, R, which is a measure of the difference in concentration of light material of the plume to its surrounding fluid, the Prandtl number, σ, which is the ratio of viscosity, ν, to thermal diffusivity, κ, the thickness of the plume, 2x0, and the distance, d, between the two vertical walls relative to the salt-finger length scale. The influence of the boundary on the fluxes of material, heat, and buoyancy is examined to find that the buoyancy flux possesses a local maximum for moderate to small thicknesses of the plume when they lie close to the wall. This has the effect of introducing a region of instability for thin plumes near the wall with an asymptotically larger growth rate. In addition, the presence of the boundary suppresses the three-dimensional instabilities present in the unbounded domain and allows only two-dimensional instabilities for moderate to small distances between the bounding walls.

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Al-Mashrafi, K. and Eltayeb, I. (2014) The Influence of Boundaries on the Stability of Compositional Plumes. Open Journal of Fluid Dynamics, 4, 83-102. doi: 10.4236/ojfd.2014.41007.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Rees Hones, D.W. and Worster, M.G. (2013) Fluxes through Steady Chimneys in a Mushy Layer during Binary Alloy Solidification. Journal of Fluid Mechanics, 714, 127-151.
[2] Wells, A.J., Wettlaufer, J.S. and Orszag, S.A. (2011) Brine Fluxes from Growing Sea Ice. Geophysical Research Letters, 38, Article ID: L04501.
[3] Loper, D.E. (1978) The Gravitationally Powered Dynamo. Geophysical Journal International, 54, 389-404.
[4] Moffatt, H.K. (1989) Liquid Metal MHD and the Geodynamo. In: Lielpeteris, J. and Moreau, R., Eds., Liquid Metal Magnetohydrodynamics, Kluwer Academic Publishers, Dordrecht, 403-412.
[5] Al-Lawatia, M.A., Elbashir, T.B.A., Eltayeb, I.A., Rahman, M.M. and Balakrishnan, E. (2011) The Dynamics of Two Interacting Compositional Plumes in the Presence of a Magnetic Field. Geophysical & Astrophysical Fluid Dynamics, 105, 586-615.
[6] Copley, S.M., Giamel, A.F., Johnson, S.M. and Hornbecker, M.F. (1970) The Origin of Freckles in Unidirectionally Solidified Casting. Metallurgical Transactions, 1, 2193-2204.
[7] Huppert, H.E. (1990) The Fluid Mechanics of Solidification. Journal of Fluid Mechanics, 212, 209-240.
[8] Chen, C.F. and Chen, F. (1991) Experimental Study of Directional Solidification of Aqueous Ammonium Chloride Solution. Journal of Fluid Mechanics, 227, 567-586.
[9] Tait, S. and Jaupart, C. (1992) Compositional Convection in a Reactive Crystalline Mush and Melt Differentiation. Journal of Geophysical Research, 97, 6735-6756.
[10] Jellinek, A.M., Kerr, R.C. and Griffiths, R.W. (1999) Mixing and Compositional Stratification Produced by Natural Convection: 1. Experiments and Their Application to Earth’s Core and Mantle. Journal of Geophysical Research, 104, 7183-7201.
[11] Classen, S., Heimpel, M. and Christensen, U. (1999) Blob Instability in Rotating Compositional Convection. Geophysical Research Letters, 26, 135-138.
[12] Aussillous, P., Sederman, A.J., Gladden, L.F., Huppert, H.E. and Worster, M.G. (2006) Magnetic Resonance Imaging of Structure and Convection in Solidifying Mushy Layers. Journal of Fluid Mechanics, 522, 99-125.
[13] Pol, S., Fernando, H.J.S. and Webb, S. (2010) Evolution of Double Diffusive Convection in Low-Aspect Ratio Containers. American Physical Society, 55, 81.
[14] Hellawell, A., Sarazin, J.R. and Steube, R.S. (1993) Channel Convection in Partly Solidified Systems. Philosophical Transactions of the Royal Society A, 345, 507-544.
[15] Eltayeb, I.A. and Loper, D.E. (1991) On the Stability of Vertical Double-Diffusive Interfaces. Part 1. A Single Plane Interface. Journal of Fluid Mechanics, 228, 149-181.
[16] Eltayeb, I.A. and Loper, D.E. (1994) On the Stability of Vertical Double-Diffusive Interfaces. Part 2. Two Parallel Interfaces. Journal of Fluid Mechanics, 267, 251-271.
[17] Eltayeb, I.A. and Loper, D.E. (1997) On the Stability of Vertical Double-Diffusive Interfaces. Part 3. Cylindrical Interfaces. Journal of Fluid Mechanics, 353, 45-66.
[18] Eltayeb, I.A. (2006) The Stability of a Compositional Plume Rotating in the Presence of a Magnetic Fluid. Geophysical & Astrophysical Fluid Dynamics, 100, 429-455.

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