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Instability due to heat diffusion in a stably stratified fluid

Published online by Cambridge University Press:  29 March 2006

Pierre Welander  and
Kjell Holmåker
Pierre Welander
Affiliation:
Oceanographic Institution, University of Gothenburg
Kjell Holmåker
Affiliation:
Department of Mathematics, Chalmers Institute of Technology, Gothenburg
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Abstract

A thermally conducting body floating in a fluid that is stably stratified by heat may develop unstable oscillations provided the body temperature shows a large enough time-lag relative to the fluid temperature. A gravitationally stable, non-Boussinesq fluid may itself become unstable in an oscillatory way through a similar diffusive time-lag. One case, that is investigated in the present work, occurs when the basic vertical temperature gradient β = dT/dz and the thermal expansion coefficient α vary in an opposite sense: β is large when α is small and vice versa. The variation in β is here assumed to be caused by internal heat sources and sinks.

If vertical oscillations are started in such a fluid temperature variations will be produced in the regions of large β but the buoyancy forces do not develop until these perturbations have diffused to regions of large α. With an appropriate lag, the buoyancy forces may give a positive work and the oscillations can grow.

Two models are investigated. The first one is a non-viscous two-layer model with β = 0, α = α0 in one layer and β = β0, α = 0 in the other layer. For this model analytical results are derived. The second model is more realistic, having continuous profiles β(z) and α(z), viscosity and horizontal boundaries. The case is studied by a numerical technique, solving the equations directly in time.

A discussion of the numerical method is given in an appendix by K. Holmaker.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 47 , Issue 1 , 14 May 1971 , pp. 51 - 64
Copyright
© 1971 Cambridge University Press

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References

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