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Double diffusive convection and steady 2D salt finger solutions in porous media

University of British Columbia

Double diffusive convection is a naturally occurring phenomenon playing important roles in geophysical, astrophysical, and oceanographic events alike. Herein, it is the transfer of heat by fluid movement driven by the differing rates of diffusion of temperature and salinity, developing into one of 2 regimes: diffusive convection or salt fingering. We consider this problem in a porous medium, relevant in situations regarding permafrost, magma, and soils amongst others. We begin by performing and comparing linear and nonlinear stability analyses near the onset of instability, as in existing work. We ultimately find that the two methods result in the same bounds for the onset of instability for salt fingering and steady diffusive convection, and so we conclude there are no subcritical cases. This is further confirmed in the third section, wherein we conduct a weakly nonlinear stability analysis using asymptotic expansions. In both the diffusive convection and the salt fingering cases, the amplitude equations obtained indicate that supercritical instabilities occur. In the case of oscillatory diffusive convection, the regime of criticality depends on the relative size of the density ratio to the Lewis number. Extending previous work by considering a porous medium, we consider modes creating the fastest growing fingers, resulting in fingers with a smaller horizontal/vertical aspect ratio. These fingers are studied first in a vertically unbounded domain, then in a bounded one. We find evolution equations for both cases, and plot the resulting steady-state solution of the temperature amplitude of the latter. We find that in the zero limit of the horizontal/vertical aspect ratio, the temperature amplitudes of the steady solutions converge in both the salt-heat and sugar-salt configurations.

Text

2020-08-25

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75689

Mathematics

University of British Columbia

UBCV

http://creativecommons.org/licenses/by-nc-nd/4.0/