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We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

Thermosolutal convection provides a testbed for applications of nonlinear dynamics to fluid motion. If the ratio of solutal to thermal diffusivity is small and the solutal Rayleigh number RS is large, instability sets in at a Hopf bifurcation as the thermal Rayleigh number RT is increased. For two-dimensional convection in a rectangular box the fundamental mode is a single roll with point symmetry about its axis. The symmetries of periodic and steady solutions form an eighth-order group with invariant subgroups that describe pure single-roll and multiroll solutions. A systematic numerical investigation reveals a rich variety of spatiotemporal behaviour in the regime where RS [Gt ] RT−RS > 0. Point symmetry is broken and there is a branch of spatially asymmetric periodic solutions. These mixed-mode oscillations lose their temporal symmetry in a subsequent bifurcation, followed eventually by a transition to chaos. The numerical experiments can be interpreted by relating the physical form of the solutions to an appropriate bifurcation structure.

Long-term variability in the ocean’s thermohaline circulation has attracted considerable attention recently in the context of past and future climate change. Drastic circulation changes are documented in paleoceanographic data and have been simulated by general circulation models of the ocean. The mechanism of spontaneous, abrupt changes in thermohaline circulation is studied here in an idealized context, using a two-dimensional Boussinesq fluid in a rectangular container, over 5 decades of Rayleigh number. When such a fluid is forced with a specified distribution of temperature and salinity at the surface - symmetric about a vertical axis - it attains a stable two-cell circulation, with the same symmetry. On the other hand, replacement of the specified salinity surface condition with an appropriate symmetric salt-flux condition leads to loss of stability of the symmetric circulation and gives rise to a new, asymmetric state. The extent of asymmetry depends on the magnitude of the thermal Rayleigh number, Ra, and on the strength of the salinity flux, y. An approximate stability curve in the y-Ra space, dividing the symmetric from the asymmetric states, is obtained numerically, and the entire range of asymmetric flows, from very slight dominance of one cell to its complete annihilation of the other cell, is explored. The physical mechanism of the pitchfork bifurcation from symmetric to asymmetric states is outlined. The effects of three other parameters of the problem are also discussed, along with implications of our results for glaciation cycles of the geological past and for interdecadal oscillations of the present ocean-atmosphere system.

The large negative Soret coefficient of 1N-LiI gives rise in a Rayleigh-Bénard experiment to a density distribution which is observed to stabilize the fluid layer for values of the Rayleigh number as large as 196 times the value of 1708 for the onset of convection in a pure fluid. The Soret transport also affects the convective heat flux. A power law relating heat flux and temperature difference is found with the same exponent as is found in pure fluids but with a lower value of the multiplicative constant. The Rayleigh number at the onset of power-law behaviour depends on the Soret coefficient. Three types of oscillation are seen: transient oscillations at onset, low frequency fluctuations at low Rayleigh number, and higher frequency oscillations similar to those observed in pure water. The intermediate state found after onset in NaCl solutions is not found in LiI.

Experiments are described in which the stability of a liquid stabilized by a gradient of concentration of a solute has been investigated. In all the cases observed, convection began at marginal stability with an oscillatory motion. The boundary of stability in the (R, Rs)-plane departs somewhat from that predicted by the simplest theories; the boundary is a curve, concave to the origin, and the critical Rayleigh number is high. The total rate of transport of solute increases much more rapidly than that of heat when the oscillations begin. The motions begin in cells which are separate from one another.

The suggestion made by the authors in a previous paper (Hurle & Jakeman 1969) that the Soret effect could give rise to overstable solutions of the thermosolutal Rayleigh–Jeffreys problem is investigated theoretically and experimentally.Oscillatory instability is shown to occur in initially homogeneous layers of water-methanol mixtures when they are heated from below. This instability triggers a finite-amplitude steady mode. The magnitude and sign of the Soret coefficient was changed by varying the composition of the mixture; as predicted, overstable modes were observed when the sign of the coefficient was such as to produce a stabilizing contribution to the density gradient. The observed critical Rayleigh numbers and temporal frequencies are consistent with theory.

Les mouvements 2D engendres par l'instabilite de circulation de Langmuir de couches stratifiees d'eau de profondeur finie sont etudies avec une hypothese simplificatrice qui les rendent analogues mathematiquement a la convection thermosolutale a double diffusion avec une concentration en solute et un flux thermique constants aux frontieres

Etude experimentale Pb-30% poids Tl en vue d'obtenir des informations significatives sur les diagrammes convectifs dans l'etat liquide adjacent au front de solidification, a partir d'une connaissance de la forme macroscopique du joint de phase. Cette forme est determinee comme une fonction du confinement lateral θ

In certain parameter regimes, it is possible to derive third-order sets of ordinary differential equations that are asymptotically exact descriptions of weakly nonlinear double convection and that exhibit chaotic behaviour. This paper presents a unified approach to deriving such models for two-dimensional convection in a horizontal layer of Boussinesq fluid with lateral constraints. Four situations are considered: thermosolutal convection, convection in an imposed vertical or horizontal magnetic field, and convection in a fluid layer rotating uniformly about a vertical axis. Thermosolutal convection and convection in an imposed horizontal magnetic field are shown here to be governed by the same sets of model equations, which exhibit the period-doubling cascades and chaotic solutions that are associated with the Shil'nikov bifurcation (Proctor & Weiss 1990). This establishes, for the first time, the existence of chaotic solutions of the equations governing two-dimensional magneto-convection. Moreover, in the limit of tall thin rolls, convection in an imposed vertical magnetic field and convection in a rotating fluid layer are both modelled by a new third-order set of ordinary differential equations, which is shown here to have chaotic solutions that are created in a homoclinic explosion, in the same manner as the chaotic solutions of the Lorenz equations. Unlike the Lorenz equations, however, this model provides an accurate description of convection in the parameter regime where the chaotic solutions appear.

Two-dimensional thermosolutal convection is perhaps the simplest example of an idealized fluid dynamical system that displays a rich variety of dynamical behaviour which is amenable to investigation by a combination of analytical and numerical techniques. The transition to chaos found in numerical experiments can be related to behaviour near a multiple bifurcation of codimension three. The resulting third-order normal form equations provide a rational approximation to the governing partial differential equations and thereby confirm that temporal chaos is present in thermosolutal convection. The complex dynamics is associated with a heteroclinic orbit in phase space linking a pair of saddle-foci with eigenvalues satisfying Shil’nikov’s criterion. The same bifurcation structure occurs in a truncated fifthorder model and numerical experiments confirm that similar behaviour extends to a significant region of parameter space.

The onset of thermosolutal convection and finite-amplitude flows, due to vertical gradients of heat and solute, in a horizontal rectangular enclosure are investigated analytically and numerically. Dirichlet or Neumann boundary conditions for temperature and solute concentration are applied to the two horizontal walls of the enclosure, while the two vertical ones are assumed impermeable and insulated. The cases of stress-free and non-slip horizontal boundaries are considered. The governing equations are solved numerically using a finite element method. To study the linear stability of the quiescent state and of the fully developed flows, a reliable numerical technique is implemented on the basis of Galerkin and finite element methods. The thresholds for finite-amplitude, oscillatory and monotonic convection instabilities are determined explicitly in terms of the governing parameters. In the diffusive mode (solute is stabilizing) it is demonstrated that overstability and subcritical convection may set in at a Rayleigh number well below the threshold of monotonic instability, when the thermal to solutal diffusivity ratio is greater than unity. In an infinite layer with rigid boundaries, the wavelength at the onset of overstability was found to be a function of the governing parameters. Analytical solutions, for finite-amplitude convection, are derived on the basis of a weak nonlinear perturbation theory for general cases and on the basis of the parallel flow approximation for a shallow enclosure subject to Neumann boundary conditions. The stability of the parallel flow solution is studied and the threshold for Hopf bifurcation is determined. For a relatively large aspect ratio enclosure, the numerical solution indicates horizontally travelling waves developing near the threshold of the oscillatory convection. Multiple confined steady and unsteady states are found to coexist. Finally, note that all the numerical solutions presented in this paper were found to be stable.

Instabilities due to a vertically stratified horizontal magnetic field (magnetic buoyancy instabilities) are believed to play a key role in the escape of the Sun's internal magnetic field and the formation of active regions and sunspots. In a star the magnetic diffusivity is much smaller than the thermal diffusivity and magnetic buoyancy instabilities are double-diffusive in character. We have studied the nonlinear development of these instabilities, in an idealized two-dimensional model, by exploiting a non-trivial transformation between the governing equations of magnetic buoyancy and those of classical thermosolutal convection. Our main result is extremely surprising. We have demonstrated the existence of finite-amplitude steady convection when both the influential gradients (magnetic and convective) are stabilizing. This strange behaviour is caused by the appearance of narrow magnetic boundary layers, which distort the mean pressure gradient so as to produce a convectively unstable stratification.

We have studied the transition between oscillatory and steady convection in a simplified model of two-dimensional thermosolutal convection. This model is exact to second order in the amplitude of the motion and is qualitatively accurate for larger amplitudes. If the ratio of the solutal diffusivity to the thermal diffusivity is sufficiently small and the solutal Rayleigh number, RS, sufficiently large, convection sets in as overstable oscillations, and these oscillations grow in amplitude as the thermal Rayleigh number, RT, is increased. In addition to this oscillatory branch, there is a branch of steady solutions that bifurcates from the static equilibrium towards lower values of RT; this subcritical branch is initially unstable but acquires stability as it turns round towards increasing values of RT. For moderate values of RS the oscillatory branch ends on the unstable (subcritical) portion of the steady branch, where the period of the oscillations becomes infinite. For larger values of RS a birfurcation from symmetrical to asymmetrical oscillations is followed by a succession of bifurcations, at each of which the period doubles, until the motion becomes aperiodic at some finite value of RT. The chaotic solutions persist as RT is further increased but eventually they lose stability and there is a transition to the stable steady branch. These results are consistent with the behaviour of solutions of the full two-dimensional problem and suggest that period-doubling, followed by the appearance of a strange attractor, is a characteristic feature of double-diffusive convection.

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.