Effect of a stabilizing gradient of solute on thermal convection

Abstract

A stabilizing gradient of solute inhibits the onset of convection in a fluid which is subjected to an adverse temperature gradient. Furthermore, the onset of instability may occur as an oscillatory motion because of the stabilizing effect of the solute. These results are obtained from linear stability theory which is reviewed briefly in the following paper before finite-amplitude results for two-dimensional flows are considered. It is found that a finite-amplitude instability may occur first for fluids with a Prandtl number somewhat smaller than unity. When the Prandtl number is equal to unity or greater, instability first sets in as an oscillatory motion which subsequently becomes unstable to disturbances which lead to steady, convecting cellular motions with larger heat flux. A solute Rayleigh number, Rs, is defined with the stabilizing solute gradient replacing the destabilizing temperature gradient in the thermal Rayleigh number. When Rs is large compared with the critical Rayleigh number of ordinary Bénard convection, the value of the Rayleigh number at which instability to finite-amplitude steady modes can set in approaches the value of Rs. Hence, asymptotically this type of instability is established when the fluid is marginally stratified. Also, as Rs → ∞ an effective diffusion coefficient, Kρ, is defined as the ratio of the vertical density flux to the density gradient evaluated at the boundary and it is found that κρ = √(κκs) where κ, κs are the diffusion coefficients for temperature and solute respectively. A study is made of the oscillatory behaviour of the fluid when the oscillations have finite amplitudes; the periods of the oscillations are found to increase with amplitude. The horizontally averaged density gradients change sign with height in the oscillating flows. Stably stratified fluid exists near the boundaries and unstably stratified fluid occupies the mid-regions for most of the oscillatory cycle. Thus the step-like behaviour of the density field which has been observed experimentally for time-dependent flows is encountered here numerically.

Convection in binary fluid mixtures. I. Extended traveling-wave and stationary states

“…All these e ects together generate the typical plume structure of the SOC concentration eld with the shafts of the plumes centered at the positions of maximal upand down ow. Such structures have also been seen in simulations of thermohaline convection 74,75]. The concentration plumes have rather narrow vertical extension and modify the concentration eld in the interior of the rolls only very slightly.…”

The diffusive regime of double-diffusive convection

“…The former seem to have a less pronounced increase in¨at low density ratios (e.g. McDougall, 1981b (Veronis, 1968;Shirtcliffe, 1973b;Linden and Shirtcliffe, 1978). At still higher density ratios, theory suggests that¨© Le § " ¢ £…”

Thermosolutal and binary fluid convection as a 2×2 matrix problem

“…In the 1960s and 1970s, Veronis [1,4] and other researchers [2,3,5,6,7,8,9,10,11,12] recognized the variety of behavior manifested by double-diffusive convection; comprehensive texts and reviews [13,14,15] were written on the subject. One of the reasons for studying these double-diffusive systems is that all display a common basic set of phenomena.…”