Double-diffusive convection, double-diffusive instabilities, density driven convection, buoyancy effects, layer formation, layer merging, flux law, pattern formation, dynamical systems, continuation methods, bifurcation analysis, stability analysis, numerical simulation
The work presented here deals with an interesting type of density driven flow called double-diffusive convection. It stems from the period while I was a Ph.D. student (1991 - 1996) at the Physics department of Utrecht University, The Netherlands.
Double-diffusive convection is an important process in oceanography and plays a role in mantle convection (magma chambers) and some technological applications. Moreover is it is an important example of multi-component density-driven convection. However, relatively few researchers are active in the field, and I hope that the results presented here may be source of inspiration and have some educational value.
Although the work is somewhat dated now (based on 2D simulations where 3D is commonplace nowadays), I believe the results still hold and are very valuable from a qualitative point of view. Also the research approach using different types of modeling (attractor & stability analysis, detailed numerical simulations, derivation of flux laws) is still appropriate today ...
The investigations reported on here are motivated by the recordings of layered structures in a stably stratified polar ocean in the vicinity of icebergs. In such an environment, a stable stratification in the ocean set up by a stabilizing salinity gradient is laterally cooled by the edge of an iceberg. The structures have been reported in several regions of both the northern part of the Antlantic and around the Antarctic. They consist of well-mixed horizontal layers with an average thickness of 5 to 15 metres, separated by thin diffusive interfaces in which steep temperature and salinity gradients exist. In the region where layers are present the overall distribution of heat and salt is stable, which indicates that lateral cooling of the ocean by the tongue is responsible for the layer formation, not vertical cooling at the top due to a cold atmosphere.
The transport of heat and salt induced by these structures is expected to be quite different from the transport that is induced by thermal convection alone, and will both influence the local transports as well as the way in which for example an iceberg melts. This motivates a deeper study on the origins of layer formation and both the qualitative and quantitative aspects of the layer formation process.
Double-diffusive convection is generated by the opposing buoyancy effects of temperature and salinity (or a couple of different fluid constituents with this property) and their unequal molecular diffusivities. Different types of motion exist depending on whether the stable stratification is provided by the component with the lowest or the highest molecular diffusivity. If the stratification is provided by the component with the lower molecular diffusivity, the stratification is of diffusive type, otherwise it is of finger type (Although finger type systems are important in oceanography, they are not discussed here). Furthermore, the direction of the gradients heavily influences the evolution of the flow.
The work presented here restricts to the case of a stable salt-stratified, initially motionless water column in a container to which a lateral temperature gradient is applied by heating or cooling (one of) the sidewalls. When the temperature difference between a vertical wall and the liquid exceeds a critical value, a layered system develops.
The first picture series at the right shows the formation of a series of layers in a square cavity of 20 cm height containing a liquid which was initially motionless and linearly stratified with salt. The salinity gradient is maintained by prescribing a constant salinity difference between the horizontal walls. The simulation was started by applying a constant lateral temperature difference by heating the left boundary of the cavity with a constant value for the temperature. As a result we end up with a double-diffusive "staircase"; a set of well-mixed convective layers separated by thin diffusive interfaces.
The next picture series shows the distibution of properties in the developed layer structure ("staircase"). These graphs correspond with the last picture of the previous series.
I am collecting information on this site, and currently this is the only page available, providing access to my Ph.D. thesis. I plan to create subpages soon with additional (hindsight) information on specific subjects. In the showcase at the right I have already made some information available, including a presentation, a unique video and a description of an experiment to create your own double-diffusive staircase in a cup of coffee ...
The chapters are available as PDF documents (some start with a blank page)
Chapter 1: Introduction and summary (PDF, 300KByte)
Summary: This chapter serves as an introduction to the research questions and results on double-diffusive convection due to sidewall heating or cooling as presented in chapters 2 -- 6 of this thesis. An example of a numerical simulation of double-diffusive convection is presented, together with a description of the mechanism of layer formation. A review of the modern literature on experimental and numerical models of double-diffusive convection due to lateral forcing then follows. Based on this review the research questions are posed that are addressed in the subsequent chapters. The contents of these chapters are briefly summarized.
Chapter 2: A bifurcation study of double-diffusive flows (PDF, 1Mbyte)
(This chapter has been published in a journal, reference  (see list below), part of the material has been published in  and )
Summary: In this chapter the double diffusive layer formation process in a laterally heated liquid layer which is stably stratified through a constant vertical salinity gradient is considered. We focus on the situation for which the salt field is fixed at the upper and lower boundaries to allow for steady state solutions. The initial layer formation, subsequent layer merging and the long time evolution are considered from a dynamical systems point of view. The structure of the stationary solutions in parameter space and their linear stability is determined using continuation methods whereas transient flows are studied through direct numerical simulation. An attempt is made to identify the boundaries between different flow regimes, as observed experimentally, as paths of particular bifurcation points in parameter space. This is only partly successful due to an abundance of singularities in some parameter regimes. However, much is learned on the dynamics of these type of flows during the attempt. For instance, the evolution towards stable states at selective points in parameter space shows that unstable steady states are physically relevant because the time at which the particular instability sets in may be very long.
Chapter 3: Layer merging during double-diffusive layer formation (PDF, 1.7Mbyte)
(This chapter has been published in a journal, reference )
Summary: The nonlinear evolution of double-diffusive instabilities into a laterally heated stably stratified motionless liquid is studied through direct numerical simulation in a two-dimensional set-up. In this chapter, we consider liquids which are initially stratified through a constant salt gradient. The stages of evolution of the intrusions and their spatial scales correspond well with those observed in laboratory experiments. A central process in the evolution is that of layer merging. A particular case of layer merging is analysed in detail and a new physical description of this process is proposed, to which we refer as 'layer sandwiching'.
Chapter 4: The physics behind self-propagating layers (PDF, 1.1Mbyte)
(This chapter has been published in a journal, reference )
Summary: Flows developing in initially doubly stratified systems are considered, i.e. in addition to a stabilizing salinity distribution a destabilizing temperature distribution is present. Lateral heating of such a system results in the formation of intrusions consisting of laterally expanding convection cells separated by diffusive interfaces. Although the development of the intrusions is qualitatively similar to that in singly stratified liquids, important differences occur when the initial destabilizing temperature gradient becomes large. When the lateral heating is turned off, intrusions are still able to propagate. The main contribution of the chapter is a detailed study of the physics of this self-propagation process.
Chapter 5: Double-diffusive layer formation near a cooled liquid-solid boundary (PDF, 1.4Mbyte)
(This chapter has been published in a journal, reference )
Summary: As an idealization of convection near an ice boundary, flows in both salt-stratified and non-stratified fluids generated by a cooled slab of solid material are considered through direct numerical simulation. When the fluid far from the slab is homogeneous, significant convection occurs below the ice and apart from a small boundary layer, hardly any flow appears next to the ice. In contrast, when the background liquid is stratified through a constant salt gradient, a layered flow appears next to the ice if the thickness of the slab is large enough. The latter flows are of double diffusive origin and have a significant effect on the transport of heat and salt near the ice.
Chapter 6: Effective diffusivities in laterally heated double-diffusive systems (PDF, 182Kbyte)
Summary: The effective vertical diffusivity for salt is estimated for laterally heated double-diffusive layered structures. First, the vertical interfacial salt fluxes are calculated using data retrieved from five different numerical simulations. The salt fluxes are shown to be consistent with the flux law for a diffusive interface, derived from laboratory experiments. The fluxes appear to be independent on the buoyancy (stability) ratio over the interface for the range considered and for a typical Rayleigh number (based on the horizontal temperature difference and vertical layer scale (h) of Rah = O(105). The effective vertical salt diffusivity KS is almost five times larger than the molecular salt diffusivity kS but is much smaller than the values available for oceanic layered structures (Rah = O(109)). An extrapolation towards Rah = O(109) yields a value of the same order as determined from measurements, namely KS= O(107) m2s-1, indicating that the vertical effective salt diffusivities are of the same order for both laterally and vertically cooled double-diffusive structures.}
Appendices (PDF, 72Kbyte: A1: The continuation technique, A2: Linear stability, A3: A flux law for vertical salt transport)
First author (Co-author: H.A. Dijkstra):
 Double-diffusive Flow Patterns in the Unicellular Flow Regime: Attractor Structure and Flow Development, in: Double-Diffusive Convection, AGU Geophysical Monograph 94, eds: Fernando & Brandt, pp. 89-96, 1995.
 The structure of (linearly) stable double diffusive flow patterns in a laterally heated stratified liquid, Phys. Fluids 7(3), pp. 680-682, 1995.
 On the evolution of double-diffusive intrusions into a stably stratified liquid: A study of the layer merging process, Int. J. Heat Mass Transfer 41, 2743-2756, 1998.
 Double diffusive layer formation near a cooled liquid-solid boundary, Int. J. Heat Mass Transfer 41, 1873-1884, 1998.
 A bifurcation study of double diffusive flows in a laterally heated stably stratified liquid layer, Int. J. Heat Mass Transfer 39, 2699-2710, 1996 (with H.A. Dijkstra).
 On the evolution of double-diffusive intrusions into a stably stratified liquid: the physics behind self-propagating layers, Int. J. Heat Mass Transfer 41, 2113-2124, 1998 (with H.A.Dijkstra).
 Layer formation in double diffusive convection (overview article), in: Time-dependent Nonlinear Convection (Advances in Fluid Mechanics), ed: P.A. Tyvand, Computational Mechanics Publications, pp. 139-176, 1998 (with H.A. Dijkstra and J. Molemaker).
Many credits go to my supervisor Henk Dijkstra (for many discussions and insights) and collegue Jeroen Molemaker (I have strongly relied on his code). Please visit Henk Dijkstra's homepage to see how he successfully combines qualitative methods from dynamical systems analysis and large-scale computational methods to obtain better understanding and thereby pushes forward research frontiers in oceanography
Contact: jurjen CURLY-A kranenborg PUNKT org