Nonlinear double-diffusive convection

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Abstract
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 R T , the saline Rayleigh number E S , 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 R S , as R T 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 R T . 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 R T 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 R T and then decreasing R T are followed. The minimum value of R T for which time-independent motion can occur is calculated for fourteen different values of σ, τ and R S . This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of σ and R S and the smaller values of τ, is less than the critical value of time-dependent linear theory.

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