Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-08T05:38:47.615Z Has data issue: false hasContentIssue false
  • English
  • Français

The effect of variable stratification on linear doubly diffusive stability

Published online by Cambridge University Press:  20 April 2006

F. Zangrando  and
Lee A. Bertram
F. Zangrando
Affiliation:
The Solar Energy Research Institute, 1617 Cole Blvd, Golden. CO 80401
Lee A. Bertram
Affiliation:
Sandia National Laboratories, Albuquerque, NM 87185
Get access Rights & Permissions [Opens in a new window]

Abstract

The effect of variable stratification on the linear bifurcations of a doubly diffusive plane parallel layer is examined numerically by expanding in a Fourier series. Because the motivation is analysis of solar-pond stability, a Prandtl number of 7 and ratio of diffusivities of $\frac{1}{80}$ is used in the study, with (large) solute Rayleigh numbers Rs ranging from 104 to 1012. Stratification of solute is cubic antisymmetric about midlayer; because temperature has a higher diffusivity, it is given a linear stratification. Exchange of stabilities results also solve the ‘fingering’ and thermal problems with cubic stratification. For the overstable case, the numerical results approach Walton's perturbation solution at large Rs, but differ significantly at smaller Rs (< 108). While both exchange of stabilities and overstable modes display an expected tendency to localize about the point of minimum solute gradient, the overstable modes behave in other, non-intuitive ways. Sublayers of reversed salinity gradient, if small enough, can be stable. Above Rs = 1012 computations become prohibitively expensive as a continuous spectrum is approached. A simple sublayer scaling rule defines an infinite family of Rs and stratification parameters on which the localized eigensolution is nearly invariant.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 151 , February 1985 , pp. 55 - 79
Copyright
© 1985 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bretherton C. S.1981 Double diffusion in a long box. Woods Hole Geophysical Fluid Dynamics Course Lectures, WHOI-81–102.Google Scholar
Busse F. H.1967 Non-stationary finite amplitude convection. J. Fluid Mech. 28, 223–239.Google Scholar
Caldwell D. R.1974 Experimental studies on the onset of thermohaline convection. J. Fluid Mech. 64, 347–367.Google Scholar
Caldwell D. R.1976 Thermosolutal convection in a solution with large negative Soret coefficient. J. Fluid Mech. 74, 129–142.Google Scholar
Chandrasekhar S.1961 Hydrodynamic and Hydromagnetic Stability. Clarendon.
Curry J. H.1978 A generalized Lorenz system. Commun. Math. Phys. 60, 193–204.Google Scholar
Da Costa, L. N., Knobloch, E. & Weiss, N. O. 1981 Oscillations in double-diffusive convection. J. Fluid Mech. 109, 25–43.Google Scholar
Gollub, J. P. & Benson S. V.1980 Time dependent instabilities and the transition to turbulent convection. J. Fluid Mech. 100, 449–470.Google Scholar
Huppert, H. E. & Linden P. F.1979 On heating a stable salinity gradient from below. J. Fluid Mech. 95, 431–464.Google Scholar
Huppert, H. E. & Moore D. R.1976 Nonlinear double diffusive convection. J. Fluid Mech. 78, 821–854.Google Scholar
Huppert, H. E. & Turner J. S.1981 Double-diffusive convection. J. Fluid Mech. 106, 229–329.Google Scholar
Joseph D. D.1976 Stability of Fluid Motions, II, p. 43. Springer.
Knobloch, E. & Proctor M. R. E.1981 Nonlinear periodic convection in double-diffusive systems. J. Fluid Mech. 108, 291–316.Google Scholar
Krishnamurti R.1968a Finite amplitude convection with changing mean temperature. Part 1. Theory. J. Fluid Mech. 33, 445–455.Google Scholar
Krishnamurti R.1968b Finite amplitude convection with changing mean temperature. Part 2. An experimental test of the theory. J. Fluid Mech. 33, 457–463.Google Scholar
Linden, P. F. & Weber J. E.1977 The formation of layers in a double-diffusive system with a sloping boundary. J. Fluid Mech. 81, 757–773.Google Scholar
Lorenz E. N.1963 Deterministic nonperiodic flow. J. Atmos. Sci. 20, 130–141.Google Scholar
Marcus P. S.1981 Effects of truncation in modal representations of thermal convection. J. Fluid Mech. 103, 241–255.Google Scholar
Proctor M. R. E.1981 Steady subcritical thermohaline convection. J. Fluid Mech. 105, 507–521.Google Scholar
Rubenfeld, L. A. & Siegmann W. L.1977 Nonlinear dynamic theory for a double-diffusive convection model. SIAM J. Appl. Maths 32, 871–894.Google Scholar
Saltzman B.1962 Finite amplitude free convection as an initial value problem. J. Atmos. Sci. 19, 329–341.Google Scholar
Sani R. L.1965 On finite amplitude roll cell disturbances in a fluid layer subjected to heat and mass transfer. AIChE J. 11, 971–980.Google Scholar
Shirtcliffe T. G. L.1967 Thermosolutal convection: observation of an overstable mode. Nature 213, 489–490.Google Scholar
Shirtcliffe T. G. L.1969 An experimental investigation of thermosolutal convection at marginal stability. J. Fluid Mech. 35, 788–688.Google Scholar
Siegmann, W. L. & Rubenfeld L. A.1975 A nonlinear model for a double-diffusive convection. SIAM J. Appl. Maths 29, 540–557.Google Scholar
Toomre J., Gough, D. O. & Spiegel E. A.1977 Numerical solutions of single-mode convection equations. J. Fluid Mech. 79, 1–31.Google Scholar
Turner J. S.1973 Buoyancy Effects in Fluids. Cambridge University Press.
Veronis G.1965 On finite amplitude instability in thermohaline convection. J. Mar. Res. 23, 1–17.Google Scholar
Veronis G.1968 Effect of a stabilizing gradient of solute on thermal convection. J. Fluid Mech. 34, 315–336.Google Scholar
Walin G.1964 Note on the stability of water stratified by both salt and heat. Tellus 16, 389–393.Google Scholar
Walton I. C.1982 Double-diffusive convection with large variable gradients. J. Fluid Mech. 125, 123–135.Google Scholar
Wright, J. H. & Loehrke R. I.1976 The onset of thermohaline convection in a linearly-stratified horizontal layer. Trans. ASME C: J. Heat Transfer 98, 558–563.Google Scholar
Zangrando F.1979 Observation and analysis of a full-scale experimental salt gradient solar pond. Ph.D. dissertation, University of New Mexico.
Zangrando, F. & Bertram L. A.1984 Doubly-diffusive linear stability with nonconstant gradients. SERI TR-252–2052.Google Scholar