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An experimental study has been carried out to examine double-diffusive convection in a porous medium. The experiments were performed in a horizontal layer of porous medium consisting of 3 mm diameter glass beads contained in a box 24 cm × 12 cm × 4 cm high. The top and bottom walls were made of brass and were kept at different constant temperatures by separate baths, with the bottom temperature higher than that of the top. The onset of convection was detected by a heat flux sensor and by the temperature distribution in the porous medium. When the porous medium was saturated with distilled water, the onset of convection was marked by a change in slope of the heat flux curve. The temperature distribution in the longitudinal direction in the middle of the layer indicated a convection pattern consisting of two-dimensional rolls with axes parallel to the short side. This pattern was confirmed by flow visualization. When the porous medium was saturated with salinity gradients of 0.15% cm−1 and 0.225% cm−1, the onset of convection was marked by a dramatic increase in heat flux at the critical ΔT, and the convection pattern was three-dimensional. When the temperature difference was reduced from supercritical to subcritical values, the heat flux curve established a hysteresis loop. Results from linear stability theory, taking into account effects of temperature-dependent viscosity, volumetric expansion coefficients, and a nonlinear basic state salinity profile, are discussed.

The partial differential equations governing two-dimensional thermosolutal convection in a Boussinesq fluid with free boundary conditions have been solved numerically in a regime where oscillatory solutions can be found. A systematic study of the transition from nonlinear periodic oscillations to temporal chaos has revealed sequences of period-doubling bifurcations. Overstability occurs if the ratio of the solutal to the thermal diffusivity tau is less than 1 and the solutal Rayleigh number Rs is sufficiently large. Solutions have been obtained for two representative values of tau. For tau = 0.316, R(s) = 10,000, symmetrical oscillations undergo a bifurcation to asymmetry, followed by a cascade of period-doubling bifurcations leading to aperiodicity, as the thermal Rayleigh number R(T) is increased. At higher values of R(T), the bifurcation sequence is repeated in reverse, restoring simple periodic solutions. As R(T) is further increased more period-doubling cascades, followed by chaos, can be identified. Within the chaotic regions there are narrow periodic windows, and multiple branches of oscillatory solutions coexist. Eventually the oscillatory branch ends and only steady solutions can be found. The development of chaos has been investigated for tau = 0.1 by varying R(T) for several different values of R(s). When R(s) is sufficiently small there are periodic solutions whose period becomes infinite at the end of the oscillatory branch. As R(s) is increased, chaos appears in the neighborhood of these heteroclinic orbits. At higher values of R(s), chaos is found for a broader range in R(T). A truncated fifth-order model suggest that the appearance of chaos is associated with heteroclinic bifurcations.

The two-dimensional motion of a fluid confined between two long horizontal planes, heated and salted from below, is examined. By a combination of perturbation analysis and direct numerical solution of the governing equations, the possible forms of large-amplitude motion are traced out as a function of the four non-dimensional parameters which specify the problem: the thermal Rayleigh number RT, the saline Rayleigh number ES, the Prandtl number σ and the ratio of the diffusivities τ. A branch of time-dependent asymptotic solutions is found which bifurcates from the linear oscillatory instability point. In general, for fixed σ, τ and RS, as RT increases three further abrupt transitions in the form of motion are found to take place independent of the initial conditions. At the first transition, a rather simple oscillatory motion changes into a more complicated one with different structure, at the second, the motion becomes aperiodic and, at the third, the only asymptotic solutions are time independent. Disordered motions are thus suppressed by increasing RT. The time-independent solutions exist on a branch which, it is conjectured, bifurcates from the time-independent linear instability point. They can occur for values of RT less than that at which the third transition point occurs. Hence for some parameter ranges two different solutions exist and a hysteresis effect occurs if solutions obtained by increasing RT and then decreasing RT are followed. The minimum value of RT for which time-independent motion can occur is calculated for fourteen different values of σ, τ and RS. This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of σ and RS and the smaller values of τ, is less than the critical value of time-dependent linear theory.

In this paper we present a rather personal view of the important developments in double-diffusive convection, a subject whose evolution has been the result of a close interaction between theoreticians, laboratory experimenters and sea-going oceano-graphers. More recently, applications in astrophysics, engineering and geology have become apparent. In the final section we attempt to draw some general conclusions and suggest that further progress will again depend on a close collaboration between fluid dynamicists and other scientists.

The onset of double‐diffusive convection is studied for an infinite fluid layer possessing a density maximum in its interior. Linear stability analyses are performed on a basic state with constant temperature and concentration gradients in which the density‐temperature relationship is quadratic. Regions of stability and instability to both steady and oscillatory modes are delineated in the positive quarterplane of the Rayleigh and solutal Rayleigh numbers plane. It is found that decreasing the position of maximum density in the vertical leads to an increased stability range and also increased regions of oscillatory instability in that quadrant. It is also found that the extent of penetration of the convective motion into the stable region is diminished with increased solute concentration. The effects of both the rigid–rigid and the free–free boundary conditions on the instabilities are investigated. Both conditions lead to dynamically similar motions with significant quantitative differences.

A model of the diffusive interface in double-diffusive convection at high Rayleigh number is proposed. The interface is assumed to have a double structure: two marginally stable boundary layers from which blobs or thermals arise on the outer edges of the interface, separated by a diffusive core across which all transport takes place by molecular diffusion. The model is time-independent and comparison is made with unsteady ‘run-down’ experiments on the assumption that the experiments run down through a sequence of equilibrium states each of which can be considered separately. The model predicts a constant ratio of the buoyancy fluxes of the two components at a value equal to the square root of the ratio of their molecular diffusivities, and individual fluxes in reasonable agreement with the available experimental data. Some time-dependent features of the model are also examined.

An asymptotic analysis of subcritical instability in double diffusive convection is presented. Using a modified perturbation method, a Landau equation that determines how the amplitude of the convection evolves in time is derived. From the Landau equation, it is found that in certain cases, stable finite amplitude convection can exist even when the rest state with no flow is locally stable. The perturbation analysis complements and unifies previous work which is primarily qualitative or numerical in character.

An analysis of double diffusive convection induced by a uniformly heated and salted horizontal wavy surface in a porous medium is presented. The wavy surface is first transformed into a smooth surface via a suitable coordinate transformation and the transformed nonsimilar coupled nonlinear parabolic equations are solved using the Keller box method. The local and average Nusselt and Sherwood numbers are given as functions of the streamwise coordinate and the effects of various physical parameters are discussed in detail. The effects of the Lewis number, buoyancy ratio, and wavy geometry on the dynamics of the flow are studied. It was found, among other observations, that the combined effect of Dufour and Soret parameters is to reduce both heat and mass transfer. MSC:34B15, 65N30, 76M20.