Nonlinear double-diffusive convection

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 RT, the saline Rayleigh number R,, the Prandtl number u and the ratio of the diffusivities r. A branch of time-dependent asymptotic solutions is found which bifurcates from the linear oscillatory instability point. In general, for fixed u, r and R,, 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 aperigdic and, at the third, the only asymptotic solutions are time independent. Disordered motions are thus suppressed by increasing R,. The time-independent solutions exist on a branch which, it is conjectured, bifurcates from the timeindependent linear instability point. They can occur for values of R, 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 u, r and Rs. This minimum value is generally much less than the critical value of time-independent linear theory and for the larger values of u and R, and the smaller values of r , is less than the critical value of time-dependent linear theory.

Homoclinic snaking of localized states in doubly diffusive convection

“…In Figure 12, this process appears to be periodic and represents a periodic attractor of Eqs. (1)- (6). Similar states have been observed in other systems as well.…”

“…(1)- (4) and boundary conditions (5)- (6). The implementation of the method follows that of Tuckerman 20 and Mamun and Tuckerman 21 and uses the first order time-stepping scheme proposed by Karniadakis et al 22 Details of the method in a three-dimensional context are described elsewhere.…”

“…The resulting SW and TW with stress-free boundaries have been studied extensively by a number of authors. 6,[16][17][18] In large domains, these waves are in turn subject to modulational instability leading to complex spatiotemporal dynamics. 19 In the present work, we are interested in dynamics in large domains with no-slip boundary conditions, focusing on spatially localized structures.…”

“…This configuration is the classic configuration in which convection sets in via overstability 6 and the resulting oscillations may coexist with strongly subcritical but steady convection. 7 Recent work of Batiste et al 8 and Bergeon and Knobloch 9 showed that steady subcritical convection favors the presence of spatially localized convective structures called convectons and revealed that such structures are present in a well-defined parameter interval called the snaking or pinning region.…”

The instability of diffusive convection and its implication for the thermohaline staircases in the deep Arctic Ocean

“…The instability of DC has been extensively studied based on the results in laboratory experiments (Veronis, 1965;Turner, 1973;Caldwell, 1974;Huppert and Moore, 1976; Pearlstein, 1981). Basically, two modes may occur at the onset of convection; one is steady convection and the other is oscillatory convection.…”