~ ) Pergamon lnt, J. Heat Mass Transfer. Vol. 39, No. 13, pp. 2699-2710, 1996 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All tights reserved 0017-9310/96 $15.00+0.00 0017-9310(95)00364-9 A bifurcation study of double diffusive flows in a laterally heated stably stratified liquid layer H. A. DIJKSTRA and E. J. KRANENBORG Institute for Marine and Atmospheric Research, Utrecht University, Utrecht, The Netherlands (Received 5 January 1995 and infinal form 26 September 1995) Abstract---The double diffusive layer formation process in a laterally heated liquid layer which is stably stratified through a constant vertical salinity gradient is considered. The salt field is fixed at the horizontal boundaries to allow for steady solutions. The structure of stationary solutions and their linear stability is determined using continuation methods. Boundaries between different flow regimes, as observed exper- imentally, are to some extent identified as paths of particular bifurcation points. Using time integrations it is shown that unstable steady-states are physically relevant because the time at which the particular instability sets in may be very long. Copyright © 1996 Elsevier Science Ltd. 1. INTRODUCTION When a lateral temperature gradient is applied to a motionless liquid layer which is stably stratified through a constant vertical salinity gradient ~b0, a buoyancy driven flow appears. This flow may become unstable when a critical value of the lateral tem- perature gradient is exceeded. The instabilities are shear driven for small q~0,but when q~0 is large the flow becomes unstable to double diffusive instabilities. The latter are due to the different thermal and solutal diffusivities. When a parcel of liquid near the heated wall moves upward, it retains almost all of its salt due to its very small diffusivity, but loses its heat relatively fast. The parcel rises to a level where its density is equal to that of the surrounding liquid and because of continuity it is then forced to move laterally; a layered flow pattern results. The vertical temperature and salinity structures associated with these layers show characteristic step- like structures. Such step structures in temperature and salinity have been found over large areas in the upper ocean. Since the presence of layers significantly influences the transport of heat and salt, double diffusive convection is a potentially important trans- port mechanism, e.g. for heat and salt in the ocean [1]. Apart from the oceanographic context, there are many technological motivations to study these type of flows, for example crystal growth and the heat storage in solar ponds. Much information on the layer formation process was obtained from laboratory experiments. These were performed either in narrow slots or in wide tanks. They differ also in the way the heating is imposed at the lateral walls, for example very slowly [2] or through a particular time dependence [3-5]. In most of the experiments, three stages of flow development are observed. There is an initial stage characterized either by spontaneous cell formation along the heated wall or by flow developing from the horizontal bound- aries. In the latter flows, the cells at the horizontal boundaries penetrate towards the center of the cavity during the second stage of evolution. In the first type of flows, layers merge during the second stage leading to an increase in the average thickness of the layers. Eventually, in both cases, a quasi-steady pattern forms with a layered structure over the whole container. Good sets of pictures showing these three stages can, for example, be found in ref. [4]. The experiments indicate that there is a boundary in parameter space separating these two qualitatively different regimes of flow. In ref. [3], a boundary in parameter space was proposed as the critical value of a Rayleigh number Ra, based on the length scale ctAT -/~o' (l) Here AT is the laterally imposed temperature difference and ~, fl the thermal and solutal expansion coefficients, respectively. The length scale r/is directly related to the movement of a heated liquid parcel to its neutrally buoyant level. It was found [3] that when Ra, exceeds a critical value given approximately by Ra~.¢r = 1.5 x 104, the layers formed simultaneously. In this case, a layered convection pattern with a ver- tical lengthscale ~/ is developed [4, 6, 7]. Below the critical value the layers grow successively from the horizontal walls, and layers with a larger scale may develop. Theoretical work has mainly focused on the initial stage of layer formation as an instability of a weak buoyancy driven background flow. In the ideal situ- ation of a vertically unbounded layer this flow is paral- lel, with liquid rising near the hot wall and descending along the cold wall. The parallel flow can be calculated analytically [8] and at large salinity gradient ~bo,it can 2699